Optimization in R n Introduction


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1 Optimization in R n Introduction Rudi Pendavingh Eindhoven Technical University Optimization in R n, lecture Rudi Pendavingh (TUE) Optimization in R n Introduction ORN / 4
2 Some optimization problems designing a cylindrical can containing liter, with minimum surface area designing a closed spatial object of volume, with minimum surface area finding the shortest path between two cities in a road network finding the shortest path between two cities in a landscape finding a maximum number of cards without a set in the SET game max{n Z x n + y n = z n for some x, y, z Z \ {,, } } Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 2 / 4
3 Optimization An optimization problem is of the form min{f (x) x S} where S is a set and f : S R is a function. S is the feasible set f : S R is the objective function if S =, then the problem is infeasible if inf{f (x) x S} =, the problem is unbounded if y S is such that f (y) = inf{f (x) x S}, then y is an optimal solution, f (y) is the optimum, then the optimum is attained by y. Solving the optimization problem means finding an optimal solution y, or deciding that the problem is infeasible, unbounded or that the optimum is not attained. Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 3 / 4
4 Constrained optimization in R n These problems are of the form So here the feasible set is Typical constraints are min{f (x) x R n satisfies constraints} S := {x R n x satisfies constraints }. inequality constraints: g(x) for some g : R n R. In particular, linear inequalities like 3x + 5x 2 + 2x n 7 equality constraints: h(x) = for some h : R n R. In particular, linear equations like 3x + 5x 2 + 2x n = 7 integrality constraints: x Z n. The entries x,..., x n R of x are called decision variables Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 4 / 4
5 Linear optimization A linear optimization problem has a linear objective and linear inequality constraints, e.g. max{cx Ax b, x R n } where A is a matrix, b a column vector, c a row vector. cx=d a a 2 c a 5 a 3 a4 Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 5 / 4
6 The Diet problem given: n foods items, m nutrients, with the jth food item containing a ij units of nutrient i per unit. Required units of nutrient i: b i, Costs per unit of food j: c j. find: amounts x j of each food j so that the intake of each nutrient is sufficient and so that total costs are minimal. This is a linear optimization problem min{ j c j x j j a ij x j b i for all i, x } or min{cx Ax b, x }. Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 6 / 4
7 Integer linear optimization Integer linear optimization is like linear optimization, but with integrality constraints: max{cx Ax b, x Z n } a 3 a4 a 5 c cx=d a 2 a Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 7 / 4
8 The Knapsack problem given: n items; weight of ith item :a i R; maximum weight b R; value of ith item c i R find: set of items X {,..., n} of total weight b and maximizing the total value. This is an integer linear optimization problem max{ c i x i a i x i b, x i for each i, x Z n } i i or max{cx ax b, x, x Z n }. Note: i X x i = Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 8 / 4
9 Convex optimization A convex optimization problem is a problem of the form min{f (x) g (x),..., g n (x), x R n } where f, g i : R n R are convex functions f(x)=d S Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 9 / 4
10 Certifying optimality Consider the linear optimization problem max{cx Ax b, x R n }. If ˆx R n is a feasible solution, then ˆx is optimal max{cx Ax b, x R n } cˆx there is no x R n such that Ax b, cx > cˆx How to show that a system of linear inequalities has no solution? Rudi Pendavingh (TUE) Optimization in R n Introduction ORN / 4
11 Fredholms Alternative Theorem Let A be an m n matrix, and let b R m. Either: (i) there exists an column vector x R n so that Ax = b; or (ii) there exists a row vector y R m so that ya = and yb =. In other words, there are x,... x n R such that a x + + a n x n = b... a m x + + a mn x n = b m or there exist y,..., y m R such that y (a x + + a n x n = b ).... y m (a m x + + a mn x n = b m ) + x + + x n =. Rudi Pendavingh (TUE) Optimization in R n Introduction ORN / 4
12 Farkas Lemma Theorem Let A be an m n matrix, and let b R m. Either: (i) there exists a column vector x R n so that Ax b; or (ii) there exists a row vector y R m so that y, ya = and yb <. A variant of this theorem is Theorem Let A be an m n matrix, and let b R m. Either: (i) there exists a column vector x R n so that Ax = b and x ; or (ii) there exists a row vector y R m so that ya and yb <. Farkas Lemma is the key to understanding linear optimization problems. Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 2 / 4
13 Kroneckers Approximation Theorem The following theorem deals with diophantine linear equations. Theorem Let A be an m n rational matrix, and let b Q m. Then either (i) there exists a column vector x Z n such that Ax = b; or (ii) there exists a row vector y Q m such that ya Z n and yb Z. To handle integer linear optimization problems, we would need a similar theorem for diophantine linear inequalities. Such a theorem is known only for very special matrices A. Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 3 / 4
14 Homework, AIMMS Weekly schedule, homework: rudi/2wo.html To download AIMMS: login: aimms password: AIMMS for TU/e Rudi Pendavingh (TUE) Optimization in R n Introduction ORN 4 / 4
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