An optimal transportation problem with import/export taxes on the boundary


 Reynold Barrett
 1 years ago
 Views:
Transcription
1 An optimal transportation problem with import/export taxes on the boundary Julián Toledo Workshop International sur les Mathématiques et l Environnement Essaouira, November Joint work with J. M. Mazón and J. D. Rossi
2 Schedule 1. Statement of the problem 2. Mass Transport Theory 3. The mass transport problem with import/export taxes: Duality 4. EvansGangbo Performance 5. Optimal Transport Plans 6. Limited importation/exportation
3 Statement of the problem
4 Business A business man (BM) produces some product in some factories in : f + gives the amount of product and its location. There are some consumers of the product in : f gives its distribution. BM wants: to transport all the mass f + to f (to satisfy the consumers) or to the boundary (export).
5 Costs and constrains The BM pays: the transport costs (given by the Euclidean distance) PLUS when a unit of mass is left on a point y : T e (y), the export taxes. BM must satisfy all the demand of the consumers. So that, he has to import product, if necessary, from the exterior, paying: T i (x), import taxes, for each unit of mass that enters from x.
6 Some freedom BM has the freedom to chose to export or import mass provided he transports all the mass in f + and covers all the mass of f.
7 MAIN GOAL The main goal here is to minimize the total cost of this operation, that is given by the transport cost PLUS export/import taxes. Given the set of transport plans { A(f +, f ) := µ M + ( ) : π 1 #µ = f + L N, π 2 #µ = f L N }, the goal is to obtain min µ A(f +,f ) x y dµ + T i d(π 1 #µ) + T e d(π 2 #µ) π i : R N R N are the projections: π 1 (x, y) := x, π 2 (x, y) := y. Given a Radon measure γ in X X, π 1 #γ is the marginal proj x (γ), and π 2 #γ is the marginal proj y (γ).
8 Equilibrium of masses The transport must verify the natural equilibrium of masses, nevertheless it is not necessary to impose f + (x) dx = f (y) dy.
9 Equilibrium of masses The transport must verify the natural equilibrium of masses, nevertheless it is not necessary to impose f + (x) dx = f (y) dy. In this transport problem two masses appear on, that are unknowns, the ones that encode the mass exported and the mass imported. Taking into account this masses we must have the equilibrium condition.
10 Conditions on the domains is assumed to be convex: we prevent the fact that the boundary of is crossed when the mass is transported inside.
11 Conditions on the domains is assumed to be convex: we prevent the fact that the boundary of is crossed when the mass is transported inside. On the support of the f ± we assume that supp(f + ) supp(f ) =.
12 Mass Transport Theory
13 Monge problem Given two measures µ, ν M(X ) (say X R N ) satisfying the mass balance condition dµ = dν, Monge s Problem consists in solving x T (x) dµ(x), inf T #µ=ν X X T : X R N are Borel functions pushing forward µ to ν: T #µ = ν, T #µ(b) = µ(t 1 (B)). X A minimizer T is called an optimal transport map of µ to ν. In the case that µ and ν represent the distribution for production and consumption of some commodity, the problem is to decide which producer should supply each consumer minimizing the total transport cost.
14 Relaxed Problem: MongeKantorovich Problem In general, the Monge problem is illposed. But...
15 Relaxed Problem: MongeKantorovich Problem In general, the Monge problem is illposed. But... Fix two measures µ, ν satisfying the mass balance condition. Let Π(µ, ν) the set of transport plans γ between µ and ν: proj x (γ) = µ, proj y (γ) = ν. The problem of finding a measure γ Π(µ, ν) which minimizes the cost functional K(γ) := x y dγ(x, y), in the set Π(µ, ν), is wellposed. X X A minimizer γ is called an optimal transport plan between µ and ν. Linearity makes the MongeKantorovich problem simpler than Monge original problem: a continuitycompactness argument guarantees the existence of an optimal transport plan.
16 Dual Problem: Kantorovich Duality Linear minimization problems admit a dual formulation: For (ϕ, ψ) L 1 (dµ) L 1 (dν), define J(ϕ, ψ) := ϕ dµ + ψ dν, X X and let Φ be the set of all measurable functions (ϕ, ψ) L 1 (dµ) L 1 (dν) satisfying Then ϕ(x) + ψ(y) x y for µ ν almost all (x, y) X X. inf K(γ) = sup J(ϕ, ψ). γ Π(µ,ν) (ϕ,ψ) Φ
17 And... KantorovichRubinstein Theorem Theorem (KantorovichRubinstein) where min K(γ) = γ Π(µ,ν) sup u K 1(X ) X u d(µ ν), K 1 (X ) := {u : X R : u(x) u(y) x y x, y X } is the set of 1Lipschitz functions in X. The maximizers u of the right hand side are called Kantorovich (transport) potentials.
18 Evans and Gangbo Approach For µ = f + L N and ν = f L N, f + and f adequate Lebesgue integrable functions, Evans and Gangbo find a Kantorovich potential as a limit, as p, of solutions to { p u p = f + f in B(0, R), u p = 0 on B(0, R). Theorem u p u K 1 () as p. u (x)(f + (x) f (x)) dx = u(x)(f + (x) f (x)) dx. max u K 1()
19 Evans and Gangbo Approach For µ = f + L N and ν = f L N, f + and f adequate Lebesgue integrable functions, Evans and Gangbo find a Kantorovich potential as a limit, as p, of solutions to { p u p = f + f in B(0, R), u p = 0 on B(0, R). Theorem u p u K 1 () as p. u (x)(f + (x) f (x)) dx = u(x)(f + (x) f (x)) dx. max u K 1() Characterization of the Kantorovich potential: There exists 0 a L () such that f + f = div(a u ) in D (). Furthermore u = 1 a.e. in the set {a > 0}.
20 Evans and Gangbo Approach Evans and Gangbo used this PDE to find a proof of the existence of an optimal transport map for the classical Monge problem, different to the first one given by Sudakov in 1979 by means of probability methods.
21 The mass transport problem with import/export taxes First approach: Duality
22 A Clever Fellow
23 A Clever Fellow A clever fellow proposes the BM to leave him the planing and offers him the following deal:
24 A Clever Fellow A clever fellow proposes the BM to leave him the planing and offers him the following deal: * to pick up the product f + (x), x, he will charge him ϕ(x), * to pick it up at x, T i (x) (paying the taxes by himself), * to leave the product at the consumer s location f (y), y, he will charge him ψ(y), and * for leaving it at y, T e (y) (paying again the taxes himself).
25 A Clever Fellow A clever fellow proposes the BM to leave him the planing and offers him the following deal: * to pick up the product f + (x), x, he will charge him ϕ(x), * to pick it up at x, T i (x) (paying the taxes by himself), * to leave the product at the consumer s location f (y), y, he will charge him ψ(y), and * for leaving it at y, T e (y) (paying again the taxes himself). * He will do all that job in such a way that: ϕ(x) + ψ(y) x y T i ϕ and T e ψ on. (assuming negative payments if necessary).
26 A Clever Fellow A clever fellow proposes the BM to leave him the planing and offers him the following deal: * to pick up the product f + (x), x, he will charge him ϕ(x), * to pick it up at x, T i (x) (paying the taxes by himself), * to leave the product at the consumer s location f (y), y, he will charge him ψ(y), and * for leaving it at y, T e (y) (paying again the taxes himself). * He will do all that job in such a way that: ϕ(x) + ψ(y) x y T i ϕ and T e ψ on. (assuming negative payments if necessary). Observe that we need to impose the following condition: T i (x) T e (y) x y, x, y, (there is not benefit of importing mass from x and exporting it to y). For x = y : T i (x) + T e (x) 0 means that in the same point the sum of exportation and importation taxes is non negative.
27 Clever Fellow s Goal Consider J(ϕ, ψ) := ϕ(x)f + (x) dx + ψ(y)f (y) dy, defined for continuous functions, and let { (ϕ, ψ) C() C() : ϕ(x) + ψ(y) x y, B(T i, T e ) := T i ϕ, T e ψ on The aim of the fellow that helps the business man is to obtain sup {J(ϕ, ψ) : (ϕ, ψ) B(T i, T e )}. }.
28 Clever Fellow s Goal Consider J(ϕ, ψ) := ϕ(x)f + (x) dx + ψ(y)f (y) dy, defined for continuous functions, and let { (ϕ, ψ) C() C() : ϕ(x) + ψ(y) x y, B(T i, T e ) := T i ϕ, T e ψ on The aim of the fellow that helps the business man is to obtain sup {J(ϕ, ψ) : (ϕ, ψ) B(T i, T e )}. Now, given (ϕ, ψ) B(T i, T e ) and µ A(f +, f ): J(ϕ, ψ) = ϕ(x)f + (x) dx + ψ(y)f (y) dy = ϕ(x)dπ 1 #µ ϕdπ 1 #µ + ψ(y)dπ 2 #µ ψdπ 2 #µ }.
29 Clever Fellow s Goal Consider J(ϕ, ψ) := ϕ(x)f + (x) dx + ψ(y)f (y) dy, defined for continuous functions, and let { (ϕ, ψ) C() C() : ϕ(x) + ψ(y) x y, B(T i, T e ) := T i ϕ, T e ψ on The aim of the fellow that helps the business man is to obtain sup {J(ϕ, ψ) : (ϕ, ψ) B(T i, T e )}. Now, given (ϕ, ψ) B(T i, T e ) and µ A(f +, f ): J(ϕ, ψ) = ϕ(x)f + (x) dx + ψ(y)f (y) dy = ϕ(x)dπ 1 #µ ϕdπ 1 #µ + ψ(y)dπ 2 #µ ψdπ 2 #µ x y dµ + T i d(π 1 #µ) + T e d(π 2 #µ). }.
30 Acceptance of the deal Therefore, sup (ϕ,ψ) B(T i,t e) inf µ A(f +,f ) J(ϕ, ψ) x y dµ + T i d(π 1 #µ) + T e d(π 2 #µ).
31 Acceptance of the deal Therefore, sup (ϕ,ψ) B(T i,t e) inf µ A(f +,f ) J(ϕ, ψ) x y dµ + T i d(π 1 #µ) + T e d(π 2 #µ). This inequality will imply that the business man accept the offer.
32 NO GAP But, in fact, there is no gap between both costs: Theorem Assume that T i and T e satisfy T i (x) T e (y) < x y x, y. Then, sup J(ϕ, ψ) (ϕ,ψ) B(T i,t e) = min µ A(f +,f ) x y dµ + T i dπ 1 #µ + T e dπ 2 #µ. Proof based on FenchelRocafellar s duality Theorem.
33 UNKNOWNS There is an important difference between the problem we are studying and the classical transport problem: there are masses, the ones that appear on the boundary, that are unknown variables.
34 UNKNOWNS There is an important difference between the problem we are studying and the classical transport problem: there are masses, the ones that appear on the boundary, that are unknown variables. The above result is an abstract result and does not give explicitly the measures involved in the problem. But...
35 EvansGangbo Performance Kantorovich potentials and optimal export/import masses
36 Let f L () and N < p < +. Let g i C( ), g 1 g 2 on. Set Wg 1,p 1,g 2 () = {u W 1,p () : g 1 u g 2 on } and consider the energy functional u(x) p Ψ p (u) := dx f (x)u(x) dx. p
37 Let f L () and N < p < +. Let g i C( ), g 1 g 2 on. Set Wg 1,p 1,g 2 () = {u W 1,p () : g 1 u g 2 on } and consider the energy functional u(x) p Ψ p (u) := dx f (x)u(x) dx. p Wg 1,p 1,g 2 () is a closed convex subset of W 1,p () and Ψ p is convex, l.s.c. and coercive. Then min Ψ p (u) u Wg 1,p 1,g 2 () has a minimizer u p in W 1,p g 1,g 2 (), which is a least energy solution of the obstacle problem { p u = f in, g 1 u g 2 on.
38 Theorem Assume g 1, g 2 satisfy g 1 (x) g 2 (y) x y x, y. Then, up to subsequence, u p u uniformly, and u is a maximizer of the variational problem { } max w(x)f (x) dx : w Wg 1, 1,g 2 (), w L () 1.
39 Theorem Assume g 1, g 2 satisfy g 1 (x) g 2 (y) x y x, y. Then, up to subsequence, u p u uniformly, and u is a maximizer of the variational problem { } max w(x)f (x) dx : w Wg 1, 1,g 2 (), w L () 1. A natural question: Let f + and f be the positive and negative part of f. Can u be interpreted as a kind of Kantorovich potential for some transport problem involving f + and f?
40 DUALITY Theorem If then g 1 (x) g 2 (x) x y x, y, u (x)(f + (x) f (x))dx = sup J(ϕ, ψ) (ϕ,ψ) B( g 1,g 2) = min µ A(f +,f ) x y dµ g 1 dπ 1 #µ + where u is the maximizer given in the above Theorem. g 2 dπ 2 #µ,
41 DUALITY Theorem If then g 1 (x) g 2 (x) x y x, y, u (x)(f + (x) f (x))dx = sup J(ϕ, ψ) (ϕ,ψ) B( g 1,g 2) = min µ A(f +,f ) x y dµ g 1 dπ 1 #µ + where u is the maximizer given in the above Theorem. g 2 dπ 2 #µ, Observe that setting T i = g 1 and T e = g 2 we are dealing with our transport problem.
42 Key result Theorem Assume g 1 (x) g 2 (y) < x y x, y. Let u p, u the functions obtained above. Then, up to a subsequence, X p := Du p p 2 Du p X weakly* in the sense of measures, div(x ) = f in the sense of distributions in, Moreover, the distributions X p η defined as X p η, ϕ := X p ϕ f ϕ for ϕ C0 (R N ), are Radon measures supported on, that, up to a subsequence, X p η V weakly* in the sense of measures; and...
43 Key result u is a Kantorovich potential for the classical transport problem of the measure ˆf + = f + L N + V + to ˆf = f L N + V. Moreover, u = g 1 on supp(v + ), u = g 2 on supp(v ).
44 Key result This result provides a proof for the previous duality Theorem: u d(ˆf + ˆf ) = min x y dν = x y dν 0, ν Π(ˆf +,ˆf ) for some ν 0 Π(ˆf +, ˆf ). Then, since π 1 #ν 0 = V + and π 2 #ν 0 = V, u (f + f ) = x y dν 0 g 1 dπ 1 #ν 0 + g 2 dπ 2 #ν 0.
45 Moreover, we have that V + and V are import and export masses in our original problem.
46 Proof Let u p be a minimizer of Ψ p (u) := u(x) p p dx f (x)u(x) dx. Then, { p u p = f in, g 1 u p g 2 on.
47 Proof Let u p be a minimizer of Ψ p (u) := u(x) p p dx f (x)u(x) dx. Then, { p u p = f in, g 1 u p g 2 on. Therefore for X p := Du p p 2 Du p we have that div(x p ) = f in the sense of distributions, and that the distribution X p η, defined as X p η, ϕ := X p ϕ f ϕ for ϕ C0 (R N ), is supported on.
48 We proof that, in fact, X p η is the sum of a nonnegative and a nonpositive distribution, so it is a measure, and moreover that supp((x p η) + ) {x : u p (x) = g 1 (x)}, and supp((x p η) ) {x : u p (x) = g 2 (x)}. In addition, we have that X p ϕ = f ϕ + ϕ d (X p η) ϕ W 1,p ().
49 Boundedness of u p, X p and X p η By using adequate test functions we get that, for p N + 1, u p p C, C constant not depending on p. d(x p η) ± C,
50 Boundedness of u p, X p and X p η By using adequate test functions we get that, for p N + 1, u p p C, d(x p η) ± C, C constant not depending on p. Therefore, there exists a sequence p i such that u pi u uniformly in, with u 1, and X pi X weakly as measures in, X pi η V weakly as measures on.
51 Then That is, formally, ϕ dx = f ϕ dx + ϕdv div(x ) = f in ϕ C 1 (). X η = V on.
52 Then That is, formally, ϕ dx = f ϕ dx + ϕdv div(x ) = f X η = V in on. ϕ C 1 (). But also we prove that fu + u dv = d X. And this allows to prove that for any 1 Lipschitz continuous function w, u f dx + u dv wf dx + wdv.
53 Giving ϕ C 1 () with ϕ 1, u f dx + u dv = = ϕ dx = ϕf dx + d X ϕdv. X ϕ d X X Then, by approximation, given a Lipschitz continuous function w with w 1, we obtain u f dx + u dv wf dx + wdv.
54 Conclusion Therefore u is a Kantorovich potential for the classical transport problem associated to the measures f + dl N + V + and f dl N + V (the mass of both measures is the same).
55 Conclusion Therefore u is a Kantorovich potential for the classical transport problem associated to the measures f + dl N + V + and f dl N + V (the mass of both measures is the same). Moreover, V + and V are import and export masses in our original problem.
56 Optimal Transport Plans
57 Detailing the transport Once export/import masses on the boundary are fixed, V + and V, take an optimal transport plan µ for f + L N + V + and f L N + V, (remark that necessarily µ( ) = 0). The part of f + exported is f + = π 1 #(µ ), and the part of f imported is f = π 2 #(µ ).
58 An optimal transport plan via transport maps. Kantorovich potential. t 2 : supp( f + ) an optimal map pushing f + forward V +. t 1 : supp( f ) an optimal map pushing f forward V. t 0 : supp(f + ) an optimal map pushing f + f + forward to f f.
59 An optimal transport plan via transport maps. Kantorovich potential. t 2 : supp( f + ) an optimal map pushing f + forward V +. t 1 : supp( f ) an optimal map pushing f forward V. t 0 : supp(f + ) an optimal map pushing f + f + forward to f f. Then, µ (x, y) = f + (x)δ y=t2(x) + (f + (x) f + (x))δ y=t0(x) + f (y)δ x=t1(y) is an optimal transport plan for our problem. u is a Kantorovich potential for each of these transports problems.
60 u (x) = min y (g 2(y) + x y ) for a.e. x supp( f + ), u (x) = max y (g 1(y) x y ) for a.e. x supp( f ).
61 T i = 0, T e = 1 2 y x
62 T i = 0, T e = 1 2 y x Importing mass from 1 is free of taxes, nevertheless, doing this would imply to export more mass to 0 and export taxes make this expensive.
63 T i = 0, T e = 1 4 y x Decreasing the export taxes, more mass is exported to 0 and some mass is imported from 1.
64 T i = 0, T e(0) = 3 4, Te(1) = 0 y x Increasing export taxes in 0, all the other null, implies that the optimal transport plan exports to 1.
65 T i = 0, T e = 1 2 y x
66 T i = 0, T e = 1 2 y x Observe that this transport problem would have coincided with the classical one if we had put T e = 1.
67 Limited importation/exportation
68 More realistic We have been using as an infinite reserve/repository.
69 More realistic We have been using as an infinite reserve/repository. We can impose the restriction of not exceeding the punctual quantity of M e (x) when we export some mass through x and also a punctual limitation of M i (x) for importing from x. We will assume M i, M e L ( ) with M i, M e 0 on.
70 More realistic We have been using as an infinite reserve/repository. We can impose the restriction of not exceeding the punctual quantity of M e (x) when we export some mass through x and also a punctual limitation of M i (x) for importing from x. We will assume M i, M e L ( ) with M i, M e 0 on. A natural constraint must be verified: M e (f + f ) that is, the interplay with the boundary is possible. Hence the mass transport problem is possible, and the problem becomes, as before, to minimize the cost. M i,
71 Especial situations 1. When moreover T i = T e = 0: limited importation/exportation but without taxes. 2. When M i (x) = 0 or M e (x) = 0 on certain zones of the boundary: it is not allowed importation or exportation in those zones. 2. M i (x) = M e (x) = 0 on the whole boundary ( f + = f ) we recover the classical MongeKantorovich mass transport problem.
72 New goal Given now A l (f +, f ) := µ M + ( ) : π 1 #µ = f + L N, π 2 #µ = f L N, π 1 #µ M i, π 2 #µ M e, we want to obtain min µ A l (f +,f ) x y dµ + T i d(π 1 #µ) + T e d(π 2 #µ). Conditions on T i and T e : T i + T e 0 on.
73 The Clever Fellow again New deal: * to pick up the product f + (x), x, he will charge him ϕ(x), * to pick it up at x, T i (x) (paying the taxes by himself) * to leave the product at the consumer s location f (y), y, he will charge him ψ(y), and * for leaving it at y, T e (y) (paying again the taxes himself).
74 The Clever Fellow again New deal: * to pick up the product f + (x), x, he will charge him ϕ(x), * to pick it up at x, T i (x) (paying the taxes by himself) * to leave the product at the consumer s location f (y), y, he will charge him ψ(y), and * for leaving it at y, T e (y) (paying again the taxes himself). * he will do all this in such a way that: ϕ(x) + ψ(y) x y T i ϕ and T e ψ on,
75 The Clever Fellow again New deal: * to pick up the product f + (x), x, he will charge him ϕ(x), * to pick it up at x, T i (x) (paying the taxes by himself) * to leave the product at the consumer s location f (y), y, he will charge him ψ(y), and * for leaving it at y, T e (y) (paying again the taxes himself). * he will do all this in such a way that: ϕ(x) + ψ(y) x y T i ϕ and T e ψ on, (now:) * he will pay a compensation when ϕ < T i or ψ < T e of: M i ( ϕ T i ) + + M e ( ψ T e ) +.
76 The aim of the fellow now is to obtain ϕf + + ψf M i ( ϕ T i ) + sup ϕ(x)+ψ(y) x y M e ( ψ T e ) +.
77 The aim of the fellow now is to obtain ϕf + + ψf M i ( ϕ T i ) + sup ϕ(x)+ψ(y) x y M e ( ψ T e ) +. And again There is no gap between both costs.
78 New energy functional The energy functional to be considered now is u(x) p Ψ l (u) := dx f (x)u(x) dx + j(x, u(x)), p M i (x)( T i (x) r) if r < T i (x), j(x, r) = 0 if T i (x) r T e (x), M e (x)(r T e (x)) if r > T e (x). For the minimizers of we prove the following result: min Ψ l(u), u W 1,p ()
79 Theorem Up to subsequence, u p u uniformly as p.
80 Theorem Up to subsequence, u p u uniformly as p. There exist X p η L ( ), X p η j(x, u p ) a.e. x such that X p η ϕ = Du p p 2 Du p ϕ f ϕ for all ϕ W 1,p (), so u p are solutions of the nonlinear boundary problem { p u = f in, u p 2 u η + j(, u( )) 0 on.
81 Theorem Up to subsequence, u p u uniformly as p. There exist X p η L ( ), X p η j(x, u p ) a.e. x such that X p η ϕ = Du p p 2 Du p ϕ f ϕ for all ϕ W 1,p (), so u p are solutions of the nonlinear boundary problem { p u = f in, u p 2 u η + j(, u( )) 0 on. X p η V weakly in L ( ) with V + M i, V M e.
82 Theorem sup w(x)f (x) dx j(x, w(x)) w W 1, (), w L () 1 = u (x)(f + (x) f (x))dx j(x, u (x)) = sup ϕ(x)+ψ(y) x y ϕf + + ψf M i ( ϕ T i ) + M e ( ψ T e ) + = min x y dµ + T i dπ 1 #µ + T e dπ 2 #µ. µ A l (f +,f )
83 u is a Kantorovich potential for the classical transport problem for the measures f + L N + V + dh N 1 and f L N + V dh N 1. V + M i and V M e are the import and export masses.
84 T i = 0, T e = 1 2, M i = M e = y x Cost: 56/16 2.
85 T i = 0, T e = 1 2, M i = M e = 1 16 y x Cost: 58/16 2.
86 Some references L. V. Kantorovich. On the tranfer of masses, Dokl. Nauk. SSSR 37 (1942), V. N. Sudakov. Geometric problems in the theory of infinitedimensional probability distributions, Proc. Stekelov Inst. Math., 141 (1979), L. C. Evans and W. Gangbo. Differential equations methods for the MongeKantorovich mass transfer problem. Mem. Amer. Math. Soc., L. C. Evans. Partial differential equations and MongeKantorovich mass transfer. Current developments in mathematics, L. Ambrosio. Lecture notes on optimal transport problems, L. Ambrosio and A. Pratelli. Existence and stability results in the L 1 theory of optimal transportation. Lecture Notes in Math., C. Villani. Topics in Optimal Transportation. Graduate Studies in Mathematics, C. Villani. Optimal Transport, Old and New. Fundamental Principles of Mathematical Sciences, 2009.
87 Shúkraan
Sur un Problème de Monge de Transport Optimal de Masse
Sur un Problème de Monge de Transport Optimal de Masse LAMFA CNRSUMR 6140 Université de Picardie Jules Verne, 80000 Amiens Limoges, 25 Mars 2010 MODE 2010, Limoges Monge problem t =t# X x Y y=t(x) Ω f
More informationA Simple Proof of Duality Theorem for MongeKantorovich Problem
A Simple Proof of Duality Theorem for MongeKantorovich Problem Toshio Mikami Hokkaido University December 14, 2004 Abstract We give a simple proof of the duality theorem for the Monge Kantorovich problem
More informationShape Optimization Problems over Classes of Convex Domains
Shape Optimization Problems over Classes of Convex Domains Giuseppe BUTTAZZO Dipartimento di Matematica Via Buonarroti, 2 56127 PISA ITALY email: buttazzo@sab.sns.it Paolo GUASONI Scuola Normale Superiore
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationQuasistatic evolution and congested transport
Quasistatic evolution and congested transport Inwon Kim Joint with Damon Alexander, Katy Craig and Yao Yao UCLA, UW Madison Hard congestion in crowd motion The following crowd motion model is proposed
More informationUNIVERSITETET I OSLO
NIVERSITETET I OSLO Det matematisknaturvitenskapelige fakultet Examination in: Trial exam Partial differential equations and Sobolev spaces I. Day of examination: November 18. 2009. Examination hours:
More informationAdaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
More informationLinear Programming Notes V Problem Transformations
Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationCHAPTER 5. Product Measures
54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue
More informationFurther Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1
Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu
COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part
More informationLinear Programming I
Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins
More informationThe integrating factor method (Sect. 2.1).
The integrating factor method (Sect. 2.1). Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Variable
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics GLOBAL EXISTENCE AND DECREASING PROPERTY OF BOUNDARY VALUES OF SOLUTIONS TO PARABOLIC EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS Sangwon Seo Volume 193 No. 1 March 2000
More informationConvex Programming Tools for Disjunctive Programs
Convex Programming Tools for Disjunctive Programs João Soares, Departamento de Matemática, Universidade de Coimbra, Portugal Abstract A Disjunctive Program (DP) is a mathematical program whose feasible
More informationNumerical Verification of Optimality Conditions in Optimal Control Problems
Numerical Verification of Optimality Conditions in Optimal Control Problems Dissertation zur Erlangung des naturwissenschaftlichen Doktorgrades der JuliusMaximiliansUniversität Würzburg vorgelegt von
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More informationFinite speed of propagation in porous media. by mass transportation methods
Finite speed of propagation in porous media by mass transportation methods José Antonio Carrillo a, Maria Pia Gualdani b, Giuseppe Toscani c a Departament de Matemàtiques  ICREA, Universitat Autònoma
More informationSensitivity analysis of utility based prices and risktolerance wealth processes
Sensitivity analysis of utility based prices and risktolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,
More informationMathematical Finance
Mathematical Finance Option Pricing under the RiskNeutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More information4.6 Linear Programming duality
4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal
More information1 Completeness of a Set of Eigenfunctions. Lecturer: Naoki Saito Scribe: Alexander Sheynis/Allen Xue. May 3, 2007. 1.1 The Neumann Boundary Condition
MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations Lecture 11: Laplacian Eigenvalue Problems for General Domains III. Completeness of a Set of Eigenfunctions and the Justification
More informationWalrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.
Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian
More informationON NONNEGATIVE SOLUTIONS OF NONLINEAR TWOPOINT BOUNDARY VALUE PROBLEMS FOR TWODIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS
ON NONNEGATIVE SOLUTIONS OF NONLINEAR TWOPOINT BOUNDARY VALUE PROBLEMS FOR TWODIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS I. KIGURADZE AND N. PARTSVANIA A. Razmadze Mathematical Institute
More informationName. Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions.
Name Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions. Question 1. A firm has a production function F (x 1, x 2 ) = ( x 1 + x 2 ) 2. It is a price
More information{f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...
44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationValuation and Optimal Decision for Perpetual American Employee Stock Options under a Constrained Viscosity Solution Framework
Valuation and Optimal Decision for Perpetual American Employee Stock Options under a Constrained Viscosity Solution Framework Quan Yuan Joint work with Shuntai Hu, Baojun Bian Email: candy5191@163.com
More informationNotes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand
Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such
More informationFuzzy Differential Systems and the New Concept of Stability
Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida
More information10. Proximal point method
L. Vandenberghe EE236C Spring 201314) 10. Proximal point method proximal point method augmented Lagrangian method MoreauYosida smoothing 101 Proximal point method a conceptual algorithm for minimizing
More informationClass Meeting # 1: Introduction to PDEs
MATH 18.152 COURSE NOTES  CLASS MEETING # 1 18.152 Introduction to PDEs, Fall 2011 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u = u(x
More informationTHE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING
THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The BlackScholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages
More informationError Bound for Classes of Polynomial Systems and its Applications: A Variational Analysis Approach
Outline Error Bound for Classes of Polynomial Systems and its Applications: A Variational Analysis Approach The University of New South Wales SPOM 2013 Joint work with V. Jeyakumar, B.S. Mordukhovich and
More informationConsumer Theory. The consumer s problem
Consumer Theory The consumer s problem 1 The Marginal Rate of Substitution (MRS) We define the MRS(x,y) as the absolute value of the slope of the line tangent to the indifference curve at point point (x,y).
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationEXISTENCE AND NONEXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION
Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 (7), pp. 5 65. ISSN: 7669. UL: http://ejde.math.txstate.edu
More informationON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction
ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZPÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that
More informationLECTURE 5: DUALITY AND SENSITIVITY ANALYSIS. 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method
LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method Introduction to dual linear program Given a constraint matrix A, right
More informationA PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of RouHuai Wang
A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of RouHuai Wang 1. Introduction In this note we consider semistable
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationEnvelope Theorem. Kevin Wainwright. Mar 22, 2004
Envelope Theorem Kevin Wainwright Mar 22, 2004 1 Maximum Value Functions A maximum (or minimum) value function is an objective function where the choice variables have been assigned their optimal values.
More information17.3.1 Follow the Perturbed Leader
CS787: Advanced Algorithms Topic: Online Learning Presenters: David He, Chris Hopman 17.3.1 Follow the Perturbed Leader 17.3.1.1 Prediction Problem Recall the prediction problem that we discussed in class.
More informationSolvability of Fractional Dirichlet Problems with Supercritical Gradient Terms.
Solvability of Fractional Dirichlet Problems with Supercritical Gradient Terms. Erwin Topp P. Universidad de Santiago de Chile Conference HJ2016, Rennes, France May 31th, 2016 joint work with Gonzalo Dávila
More informationBig Data  Lecture 1 Optimization reminders
Big Data  Lecture 1 Optimization reminders S. Gadat Toulouse, Octobre 2014 Big Data  Lecture 1 Optimization reminders S. Gadat Toulouse, Octobre 2014 Schedule Introduction Major issues Examples Mathematics
More informationL. Campi and M. Del Vigna Weak Insider Trading and Behavioral Finance
WEAK INSIDER TRADING AND BEHAVIORAL FINANCE L. Campi 1 M. Del Vigna 2 1 Université Paris XIII 2 Department of Mathematics for Economic Decisions University of Florence 5 th FlorenceRitsumeikan Workshop
More informationChapter 21: The Discounted Utility Model
Chapter 21: The Discounted Utility Model 21.1: Introduction This is an important chapter in that it introduces, and explores the implications of, an empirically relevant utility function representing intertemporal
More informationChapter 5. Banach Spaces
9 Chapter 5 Banach Spaces Many linear equations may be formulated in terms of a suitable linear operator acting on a Banach space. In this chapter, we study Banach spaces and linear operators acting on
More informationMath 5311 Gateaux differentials and Frechet derivatives
Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.
More informationMultivariable Calculus and Optimization
Multivariable Calculus and Optimization Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Multivariable Calculus and Optimization 1 / 51 EC2040 Topic 3  Multivariable Calculus
More informationGeneral Equilibrium Theory: Examples
General Equilibrium Theory: Examples 3 examples of GE: pure exchange (Edgeworth box) 1 producer  1 consumer several producers and an example illustrating the limits of the partial equilibrium approach
More informationLecture 3: Linear Programming Relaxations and Rounding
Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can
More informationMicroeconomic Theory: Basic Math Concepts
Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts
More informationMathematics for Econometrics, Fourth Edition
Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents
More informationLecture 4: Equality Constrained Optimization. Tianxi Wang
Lecture 4: Equality Constrained Optimization Tianxi Wang wangt@essex.ac.uk 2.1 Lagrange Multiplier Technique (a) Classical Programming max f(x 1, x 2,..., x n ) objective function where x 1, x 2,..., x
More informationMoral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania
Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 PrincipalAgent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically
More informationExample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x
Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written
More informationECG590I Asset Pricing. Lecture 2: Present Value 1
ECG59I Asset Pricing. Lecture 2: Present Value 1 2 Present Value If you have to decide between receiving 1$ now or 1$ one year from now, then you would rather have your money now. If you have to decide
More informationStochastic Inventory Control
Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the
More informationMATH1231 Algebra, 2015 Chapter 7: Linear maps
MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter
More informationON A MIXED SUMDIFFERENCE EQUATION OF VOLTERRAFREDHOLM TYPE. 1. Introduction
SARAJEVO JOURNAL OF MATHEMATICS Vol.5 (17) (2009), 55 62 ON A MIXED SUMDIFFERENCE EQUATION OF VOLTERRAFREDHOLM TYPE B.. PACHPATTE Abstract. The main objective of this paper is to study some basic properties
More informationFixed Point Theorems
Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationMultigrid preconditioning for nonlinear (degenerate) parabolic equations with application to monument degradation
Multigrid preconditioning for nonlinear (degenerate) parabolic equations with application to monument degradation M. Donatelli 1 M. Semplice S. SerraCapizzano 1 1 Department of Science and High Technology
More informationDIVISORS IN A DEDEKIND DOMAIN. Javier Cilleruelo and Jorge JiménezUrroz. 1 Introduction
DIVISORS IN A DEDEKIND DOMAIN Javier Cilleruelo and Jorge JiménezUrroz 1 Introduction Let A be a Dedekind domain in which we can define a notion of distance. We are interested in the number of divisors
More informationA STABILITY RESULT ON MUCKENHOUPT S WEIGHTS
Publicacions Matemàtiques, Vol 42 (998), 53 63. A STABILITY RESULT ON MUCKENHOUPT S WEIGHTS Juha Kinnunen Abstract We prove that Muckenhoupt s A weights satisfy a reverse Hölder inequality with an explicit
More informationArrangements And Duality
Arrangements And Duality 3.1 Introduction 3 Point configurations are tbe most basic structure we study in computational geometry. But what about configurations of more complicated shapes? For example,
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationSome stability results of parameter identification in a jump diffusion model
Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Abstract In this paper we discuss
More informationLecture Notes on Elasticity of Substitution
Lecture Notes on Elasticity of Substitution Ted Bergstrom, UCSB Economics 20A October 26, 205 Today s featured guest is the elasticity of substitution. Elasticity of a function of a single variable Before
More informationCONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian. Pasquale Candito and Giuseppina D Aguí
Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian Pasquale
More informationF. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p
More informationGeometrical Characterization of RNoperators between Locally Convex Vector Spaces
Geometrical Characterization of RNoperators between Locally Convex Vector Spaces OLEG REINOV St. Petersburg State University Dept. of Mathematics and Mechanics Universitetskii pr. 28, 198504 St, Petersburg
More informationLimits and Continuity
Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function
More informationExtremal equilibria for reaction diffusion equations in bounded domains and applications.
Extremal equilibria for reaction diffusion equations in bounded domains and applications. Aníbal RodríguezBernal Alejandro VidalLópez Departamento de Matemática Aplicada Universidad Complutense de Madrid,
More informationExistence de solutions à support compact pour un problème elliptique quasilinéaire
Existence de solutions à support compact pour un problème elliptique quasilinéaire Jacques Giacomoni, Habib Mâagli, Paul Sauvy Université de Pau et des Pays de l Adour, L.M.A.P. Faculté de sciences de
More informationTHE BANACH CONTRACTION PRINCIPLE. Contents
THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,
More informationDate: April 12, 2001. Contents
2 Lagrange Multipliers Date: April 12, 2001 Contents 2.1. Introduction to Lagrange Multipliers......... p. 2 2.2. Enhanced Fritz John Optimality Conditions...... p. 12 2.3. Informative Lagrange Multipliers...........
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationIncreasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.
1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.
More informationOn Integer Additive SetIndexers of Graphs
On Integer Additive SetIndexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A setindexer of a graph G is an injective setvalued function f : V (G) 2 X such that
More information15 Kuhn Tucker conditions
5 Kuhn Tucker conditions Consider a version of the consumer problem in which quasilinear utility x 2 + 4 x 2 is maximised subject to x +x 2 =. Mechanically applying the Lagrange multiplier/common slopes
More informationALMOST COMMON PRIORS 1. INTRODUCTION
ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type
More informationMathematical finance and linear programming (optimization)
Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may
More informationLecture 2: Consumer Theory
Lecture 2: Consumer Theory Preferences and Utility Utility Maximization (the primal problem) Expenditure Minimization (the dual) First we explore how consumers preferences give rise to a utility fct which
More informationCONTRIBUTIONS TO ZERO SUM PROBLEMS
CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg
More informationEconomics 121b: Intermediate Microeconomics Problem Set 2 1/20/10
Dirk Bergemann Department of Economics Yale University s by Olga Timoshenko Economics 121b: Intermediate Microeconomics Problem Set 2 1/20/10 This problem set is due on Wednesday, 1/27/10. Preliminary
More informationAbout the inverse football pool problem for 9 games 1
Seventh International Workshop on Optimal Codes and Related Topics September 61, 013, Albena, Bulgaria pp. 15133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute
More informationOn Lexicographic (Dictionary) Preference
MICROECONOMICS LECTURE SUPPLEMENTS Hajime Miyazaki File Name: lexico95.usc/lexico99.dok DEPARTMENT OF ECONOMICS OHIO STATE UNIVERSITY Fall 993/994/995 Miyazaki.@osu.edu On Lexicographic (Dictionary) Preference
More informationModern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh
Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem
More information5 Numerical Differentiation
D. Levy 5 Numerical Differentiation 5. Basic Concepts This chapter deals with numerical approximations of derivatives. The first questions that comes up to mind is: why do we need to approximate derivatives
More informationConvex analysis and profit/cost/support functions
CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m
More informationHydrodynamic Limits of Randomized Load Balancing Networks
Hydrodynamic Limits of Randomized Load Balancing Networks Kavita Ramanan and Mohammadreza Aghajani Brown University Stochastic Networks and Stochastic Geometry a conference in honour of François Baccelli
More informationSample Midterm Solutions
Sample Midterm Solutions Instructions: Please answer both questions. You should show your working and calculations for each applicable problem. Correct answers without working will get you relatively few
More informationCost Minimization and the Cost Function
Cost Minimization and the Cost Function Juan Manuel Puerta October 5, 2009 So far we focused on profit maximization, we could look at a different problem, that is the cost minimization problem. This is
More information