Convex analysis and profit/cost/support functions


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1 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 Convex analysts may give one of two definitions for the support function of A as either an infimum or a supremum Recall that the supremum of a set of real numbers is its least upper bound and the infimum is its greatest lower bound If A has no upper bound, then by convention sup A = and if A has no lower bound, then inf A = For the empty set, sup A = and inf A = ; otherwise inf A sup A This makes a kind of sense: Every real number is an upper bound for the empty set, since there is no member of the empty set that is greater than Thus the least upper bound must be Similarly, every real number is also a lower bound, so the infimum is Thus support functions as infima or suprema may assume the values and By convention, 0 = 0; if > 0 is a real number, then = and = ; and if < 0 is a real number, then = and = These conventions are used to simplify statements involving positive homogeneity Rather than choose one definition, I shall give the two definitions different names based on their economic interpretation Profit maximization The profit function π A of A is defined by π A p = sup p y y A Cost minimization The cost function c A of A is defined by c A p = inf y A p y Clearly, π A p = c A p Clearly c A p = π A p π A is convex, lower semicontinuous, and positively homogeneous of degree 1 Positive homogeneity of π A is obvious given the conventions on multiplication of infinities To see that it is convex, let g x be the linear hence convex function defined by g x p = x p Then π A p = sup x A g x p Since the pointwise supremum of a family of convex functions is convex, π A is convex Also each g x is continuous, hence lower semicontinuous, and the supremum of a family of lower semicontinuous functions is lower semicontinuous See my notes on maximization c A is concave, upper semicontinuous, and positively homogeneous of degree 1 Positive homogeneity of c A is obvious given the conventions on multiplication of infinities To see that it is concave, let g x be the linear hence concave function defined by g x p = x p Then c A p = inf x A g x p Since the pointwise infimum of a family of concave functions is concave, c A is concave Also each g x is continuous, hence upper semicontinuous, and the infimum of a family of upper semicontinuous functions is upper semicontinuous See my notes on maximization 1
2 KC Border Convex analysis and profit/cost/support functions 2 The set The set {p R m : π A p < } is a closed convex cone, called the effective domain of π A, and denoted dom π A The effective domain will always include the point 0 provided A is nonempty By convention π p = for all p, and we say that π A is improper If A = R m, then 0 is the only point in the effective domain of π A It is easy to see that the effective domain dom π A of π A is a cone, that is, if p dom π A, then p dom π A for every 0 Note that {0} is a degenerate cone It is also straightforward to show that dom π A is convex For if π A p < and π A q <, for 0 1, by convexity of π A, we π A x + 1 y πa p + 1 π A q < The closedness of dom π A is more difficult {p R m : c A p > } is a closed convex cone, called the effective domain of c A, and denoted dom c A The effective domain will always include the point 0 provided A is nonempty By convention c p = for all p, and we say that c is improper If A = R m, then 0 is the only point in the effective domain of c A It is easy to see that the effective domain dom c A of c A is a cone, that is, if p dom c A, then p dom c A for every 0 Note that {0} is a degenerate cone It is also straightforward to show that dom c A is convex For if c A p > and c A q >, for 0 1, by concavity of c A, we c A x + 1 y ca p + 1 c A q > The closedness of dom c A is more difficult Recoverability Separating Hyperplane Theorem If A is a closed convex set, and x does not belong to A, then there is a nonzero p satisfying p x > π A p For a proof see my notes Note that given our conventions, A may be empty From this theorem we easily get the next proposition The closed convex hull co A of co A = { y R m : p R m [ p y π A p ]} Separating Hyperplane Theorem If A is a closed convex set, and x does not belong to A, then there is a nonzero p satisfying p x < c A p For a proof see my notes Note that given our conventions, A may be empty From this theorem we easily get the next proposition The closed convex hull co A of co A = { y R m : p R m [ p y c A p ]} Now let f be a continuous realvalued function defined on a closed convex cone D We can extend f to all of R m by setting it to outside of D if f is convex or if f is concave If f is convex and positively homogeneous of degree 1, define A = { y R m : p R m [ p y fp ]} Then A is closed and convex and f = π A If f is concave and positively homogeneous of degree 1, define A = { y R m : p R m [ p y fp ]} Then A is closed and convex and f = c A
3 KC Border Convex analysis and profit/cost/support functions 3 Extremizers are subgradients If ỹp maximizes p over A, that is, if ỹp belongs to A and p ỹp p y for all y A, then ỹp is a subgradient of π A at p That is, If ŷp minimizes p over A, that is, if ŷp belongs to A and p ŷp p y for all y A, then ŷp is a supergradient of c A at p That is, π A p + ỹp q p π A q c A p + ŷp q p c A q for all q R m for all q R m To see this, note that for any q R m, by definition we q ỹp π A q Now add π A p p ỹp = 0 to the left hand side to get the subgradient inequality Note that π A p may be finite for a closed convex set A, and yet there may be no maximizer For instance, let A = {x, y R 2 : x < 0, y < 0, xy 1} Then for p = 1, 0, we π A p = 0 as 1, 0 1/n, n = 1/n, but 1, 0 x, y = x < 0 for each x, y A Thus there is no maximizer in A It turns out that if there is no maximizer of p, then π A has no subgradient at p In fact, the following is true, but I won t present the proof, which relies on the Separating Hyperplane Theorem See my notes for a proof Theorem If A is closed and convex, then x is a subgradient of π A at p if and only if x A and x maximizes p over A To see this, note that for any q R m, by definition we q ŷp c A q Now add c A p p ŷp = 0 to the left hand side to get the supergradient inequality Note that x A p may be finite for a closed convex set A, and yet there may be no minimizer For instance, let A = {x, y R 2 : x > 0, y > 0, xy 1} Then for p = 1, 0, we π A p = 0 as 1, 0 1/n, n = 1/n, but 1, 0 x, y = x > 0 for each x, y A Thus there is no minimizer in A It turns out that if there is no minimizer of p, then c A has no supergradient at p In fact, the following is true, but I won t present the proof, which relies on the Separating Hyperplane Theorem See my notes for a proof Theorem If A is closed and convex, then x is a supergradient of c A at p if and only if x A and x minimizes p over A Comparative statics Consequently, if A is closed and convex, and ỹp is the unique maximizer of p over A, then π A is differentiable at p and Consequently, if A is closed and convex, and ŷp is the unique minimizer of p over A, then c A is differentiable at p and ỹp = π Ap ŷp = c Ap
4 KC Border Convex analysis and profit/cost/support functions 4 To see that differentiability of π A implies the profit maximizer is unique, consider q of the form p ± e i, where e i is the i th unit coordinate vector, and > 0 The subgradient inequality for q = p + e i is ỹp e i π A p + e i π A p and for q = p e i is ỹp e i π A p e i π A p Dividing these by and respectively yields To see that differentiability of c A implies the cost minimizer is unique, consider q of the form p ± e i, where e i is the i th unit coordinate vector, and > 0 The supergradient inequality for q = p + e i is ŷp e i c A p + e i c A p and for q = p e i is ŷp e i c A p e i c A p Dividing these by and respectively yields so yi p π Ap + e i π A p yi p π Ap e i π A p so yi p c Ap + e i c A p yi p c Ap e i c A p π A p e i π A p y i p π Ap+e i π A p Letting 0 yields ỹ i p = D i π A p Thus if π A is twice differentiable at p, that is, if the maximizer ỹp is differentiable with respect to p, then the i th component D j ỹ i p = D ij π A p Consequently, the matrix [ Dj ỹ i p ] c A p+e i c A p y i p c Ap e i c A p Letting 0 yields ŷ i p = D i c A p Thus if c A is twice differentiable at p, that is, if the minimizer ŷp is differentiable with respect to p, then the i th component D j ŷ i p = D ij c A p Consequently, the matrix [ Dj ŷ i p ] is positive semidefinite is negative semidefinite In particular, D i ỹ i 0 In particular, D i ŷ i 0 Even without twice differentiability, from the subgradient inequality, we π A p + ỹp q p π A q π A q + ỹq p q π A p so adding the two inequalities, we get ỹp ỹq p q 0 Even without twice differentiability, from the supergradient inequality, we c A p + ŷp q p c A q c A q + ŷq p q c A p so adding the two inequalities, we get ŷp ŷq p q 0
5 KC Border Convex analysis and profit/cost/support functions 5 Thus if q differs from p only in its i th component, say q i = p i + p i, then we ỹ i p i 0 Dividing by the positive quantity p i 2 does not change this inequality, so ỹ i p i 0 Thus if q differs from p only in its i th component, say q i = p i + p i, then we ŷ i p i 0 Dividing by the positive quantity p i 2 does not change this inequality, so ŷ i p i 0 Cyclical monotonicity and empirical restrictions A real function g : X R R is increasing if x y = gx gy That is, gx gy and x y the same sign An equivalent way to say this is gx gy x y 0 for all x, y, which can be rewritten as gxy x + gyx y 0 for all x, y We can generalize this to a function g from R m into R m like this: Definition A function g : X R m R m is monotone increasing if gx y x + gy x y 0 for all x, y X We already seen that the subgradient of π A is monotone increasing More is true Definition A mapping g : X R m R m is cyclically monotone increasing if for every cycle x 0, x 1,, x n, x n+1 = x 0 in X, we gx 0 x 1 x gx n x n+1 x n 0 If f : X R m R is convex, and g : X R m R m is a selection from the subdifferential of f, that is, if gx is a subgradient of f at x for every x, then g is cyclically monotone increasing A real function g : X R R is decreasing if x y = gx gy That is, gx gy and x y the opposite sign An equivalent way to say this is gx gy x y 0 for all x, y, which can be rewritten as gxy x + gyx y 0 for all x, y We can generalize this to a function g from R m into R m like this: Definition A function g : X R m R m is monotone decreasing if gx y x + gy x y 0 for all x, y X We already seen that the supergradient of c A is monotone decreasing More is true Definition A mapping g : X R m R m is cyclically monotone decreasing if for every cycle x 0, x 1,, x n, x n+1 = x 0 in X, we gx 0 x 1 x gx n x n+1 x n 0 If f : X R m R is concave, and g : X R m R m is a selection from the superdifferential of f, that is, if gx is a supergradient of f at x for every x, then g is cyclically monotone decreasing
6 KC Border Convex analysis and profit/cost/support functions 6 Proof : Let x 0, x 1,, x n, x n+1 = x 0 be a cycle in X From the subgradient inequality at x i, we or fx i + gx i x i+1 x i fx i+1 gx i x i+1 x i fx i+1 fx i for each i = 0,, n Summing gives n gx i x i+1 x i 0, i=0 where the righthand side takes into account fx n+1 = fx 0 Corollary The profit maximizing points correspondence ŷ is cyclically monotonic increasing Remarkably the converse is true Theorem Rockafellar Let X be a convex set in R m and let φ: X R m be a cyclically monotone increasing correspondence that is, if every selection from φ is a cyclically monotone increasing function Then there is a lower semicontinuous convex function f : X R such that φx fx for every x Proof : Let x 0, x 1,, x n, x n+1 = x 0 be a cycle in X From the supergradient inequality at x i, we or fx i + gx i x i+1 x i fx i+1 gx i x i+1 x i fx i+1 fx i for each i = 0,, n Summing gives n gx i x i+1 x i 0, i=0 where the righthand side takes into account fx n+1 = fx 0 Corollary The cost minimizing points correspondence ŷ is cyclically monotonic decreasing Remarkably the converse is true Theorem Rockafellar Let X be a convex set in R m and let φ: X R m be a cyclically monotone decreasing correspondence that is, if every selection from φ is a cyclically monotone decreasing function Then there is an upper semicontinuous concave function f : X R such that φx fx for every x
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