linear programming.pdf - 2 Convex and Linear Optimization 2.1 Convexity and Strong Duality Let S \u2286 Rn S is called a convex set if for all \u03b4 \u2208[0 1 x

linear programming.pdf - 2 Convex and Linear Optimization...

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2 Convex and Linear Optimization 2.1 Convexity and Strong Duality Let S R n . S is called a convex set if for all δ [ 0, 1 ] , x, y S implies that δx +( 1 - δ ) y S . A function f : S R is called convex function if for all x, y S and δ [ 0, 1 ] , δf ( x ) + ( 1 - δ ) f ( y ) greaterorequalslant f ( δx + ( 1 - δ ) y ) . A point x S is called an extreme point of S if for all y, z S and δ ( 0, 1 ) , x = δy +( 1 - δ ) z implies that x = y = z . A point x S is called an interior point of S if there exists ǫ > 0 such that { y : || y - x || 2 lessorequalslant ǫ } S . The set of all interior points of S is called the interior of S . We saw in the previous lecture that strong duality is equivalent to the existence of a supporting hyperplane. The following result establishes a sufficient condition for the latter. Theorem 2.1 (Supporting Hyperplane Theorem) . Suppose that φ is convex and b R lies in the interior of the set of points where φ is finite. Then there exists a (non-vertical) supporting hyperplane to φ at b . The following result identifies a condition that guarantees convexity of φ . Theorem 2.2 . Consider the optimization problem to minimize f ( x ) subject to h ( x ) lessorequalslant b x X, and let φ be given by φ ( b ) = inf x X ( b ) f ( x ) . Then, φ is convex when X , f , and h are convex. Proof. Consider b 1 , b 2 R m such that φ ( b 1 ) and φ ( b 2 ) are defined, and let δ [ 0, 1 ] and b = δb 1 + ( 1 - δ ) b 2 . Further consider x 1 X ( b 1 ) , x 2 X ( b 2 ) , and let x = δx 1 +( 1 - δ ) x 2 . Then convexity of X implies that x X , and convexity of h that h ( x )= h ( δx 1 +( 1 - δ ) x 2 ) lessorequalslant δh ( x 1 )+( 1 - δ ) h ( x 2 ) = δb 1 +( 1 - δ ) b 2 = b. Thus x X ( b ) , and by convexity of f , φ ( b ) lessorequalslant f ( x )= f ( δx 1 +( 1 - δ ) x 2 ) lessorequalslant δf ( x 1 )+( 1 - δ ) f ( x 2 ) . This holds for all x 1 X ( b 1 ) and x 2 X ( b 2 ) , so taking infima on the right hand side yields φ ( b ) lessorequalslant δφ ( b 1 )+( 1 - δ ) φ ( b 2 ) . 7
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8 2 · Convex and Linear Optimization Observe that an equality constraint h ( x ) = b is equivalent to constraints h ( x ) lessorequalslant b and - h ( x ) lessorequalslant - b . In this case, the above result requires that X , f , h , and - h are all convex, which in particular requires that h is linear. 2.2 Linear Programs A linear program is an optimization problem in which the objective and all constraints are linear. It has the form minimize c T x subject to a T i x greaterorequalslant b i , i M 1 a T i x lessorequalslant b i , i M 2 a T i x = b i , i M 3 x j greaterorequalslant 0, j N 1 x j lessorequalslant 0, j N 2 where c R n is a cost vector, x R n is a vector of decision variables, and constraints are given by a i R n and b i R for i { 1, . . . , m } . Index sets M 1 , M 2 , M 3 { 1, . . . , m } and N 1 , N 2 { 1, . . . , n
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