v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
290 CHAPTER 4. SEMIDEFINITE PROGRAMMINGP˜PdualitydualityDtransformation˜DFigure 82: Connectivity indicates paths between particular primal and dualproblems from Exercise 4.2.2.1.1. More generally, any path between primalproblems P (and equivalent ˜P) and dual D (and equivalent ˜D) is possible:implying, any given path is not necessarily circuital; dual of a dual problemis not necessarily stated in precisely same manner as corresponding primalconvex problem, in other words, although its solution set is equivalent towithin some transformation.where S ⋆ , y ⋆ denote a dual optimal solution. 4.13 We summarize this:4.2.3.0.1 Corollary. Optimality and strong duality. [359,3.1][391,1.3.8] For semidefinite programs (649P) and (649D), assume primaland dual feasible sets A ∩ S n + ⊂ S n and D ∗ ⊂ S n × R m (661) are nonempty.ThenX ⋆ is optimal for (649P)S ⋆ , y ⋆ are optimal for (649D)duality gap 〈C,X ⋆ 〉−〈b, y ⋆ 〉 is 0if and only ifi) ∃X ∈ A ∩ int S n + or ∃y ∈ rel int ˜D ∗ii) 〈S ⋆ , X ⋆ 〉 = 0and⋄4.13 Optimality condition 〈S ⋆ , X ⋆ 〉=0 is called a complementary slackness condition, inkeeping with linear programming tradition [94], that forbids dual inequalities in (649) tosimultaneously hold strictly. [306,4]
4.2. FRAMEWORK 291For symmetric positive semidefinite matrices, requirement ii is equivalentto the complementarity (A.7.4)〈S ⋆ , X ⋆ 〉 = 0 ⇔ S ⋆ X ⋆ = X ⋆ S ⋆ = 0 (675)Commutativity of diagonalizable matrices is a necessary and sufficientcondition [202,1.3.12] for these two optimal symmetric matrices to besimultaneously diagonalizable. ThereforerankX ⋆ + rankS ⋆ ≤ n (676)Proof. To see that, the product of symmetric optimal matricesX ⋆ , S ⋆ ∈ S n must itself be symmetric because of commutativity. (1474) Thesymmetric product has diagonalization [11, cor.2.11]S ⋆ X ⋆ = X ⋆ S ⋆ = QΛ S ⋆Λ X ⋆Q T = 0 ⇔ Λ X ⋆Λ S ⋆ = 0 (677)where Q is an orthogonal matrix. The product of the nonnegative diagonal Λmatrices can be 0 if their main diagonal zeros are complementary or coincide.Due only to symmetry, rankX ⋆ = rank Λ X ⋆ and rankS ⋆ = rank Λ S ⋆ forthese optimal primal and dual solutions. (1460) So, because of thecomplementarity, the total number of nonzero diagonal entries from both Λcannot exceed n .When equality is attained in (676)rankX ⋆ + rankS ⋆ = n (678)there are no coinciding main diagonal zeros in Λ X ⋆Λ S ⋆ , and so we have whatis called strict complementarity. 4.14 Logically it follows that a necessary andsufficient condition for strict complementarity of an optimal primal and dualsolution isX ⋆ + S ⋆ ≻ 0 (679)4.2.3.1 solving primal problem via dualThe beauty of Corollary 4.2.3.0.1 is its conjugacy; id est, one can solve eitherthe primal or dual problem in (649) and then find a solution to the othervia the optimality conditions. When a dual optimal solution is known,for example, a primal optimal solution is any primal feasible solution inhyperplane {X | 〈S ⋆ , X〉=0}.4.14 distinct from maximal complementarity (4.1.2).
- Page 239 and 240: 3.4. AFFINE FUNCTION 239f(z)Az 2z 1
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- Page 257 and 258: 3.6. GRADIENT 257f(Y )[ ∇f(X)−1
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- Page 275 and 276: 4.1. CONIC PROBLEM 275(confer p.162
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4.2. FRAMEWORK 291For symmetric positive semidefinite matrices, requirement ii is equivalentto the complementarity (A.7.4)〈S ⋆ , X ⋆ 〉 = 0 ⇔ S ⋆ X ⋆ = X ⋆ S ⋆ = 0 (675)Commutativity of diagonalizable matrices is a necessary and sufficientcondition [202,1.3.12] for these two optimal symmetric matrices to besimultaneously diagonalizable. ThereforerankX ⋆ + rankS ⋆ ≤ n (676)Proof. To see that, the product of symmetric optimal matricesX ⋆ , S ⋆ ∈ S n must itself be symmetric because of commutativity. (1474) Thesymmetric product has diagonalization [11, cor.2.11]S ⋆ X ⋆ = X ⋆ S ⋆ = QΛ S ⋆Λ X ⋆Q T = 0 ⇔ Λ X ⋆Λ S ⋆ = 0 (677)where Q is an orthogonal matrix. The product of the nonnegative diagonal Λmatrices can be 0 if their main diagonal zeros are complementary or coincide.Due only to symmetry, rankX ⋆ = rank Λ X ⋆ and rankS ⋆ = rank Λ S ⋆ forthese optimal primal and dual solutions. (1460) So, because of thecomplementarity, the total number of nonzero diagonal entries from both Λcannot exceed n .When equality is attained in (676)rankX ⋆ + rankS ⋆ = n (678)there are no coinciding main diagonal zeros in Λ X ⋆Λ S ⋆ , and so we have whatis called strict complementarity. 4.14 Logically it follows that a necessary andsufficient condition for strict complementarity of an optimal primal and dualsolution isX ⋆ + S ⋆ ≻ 0 (679)4.2.3.1 solving primal problem via dualThe beauty of Corollary 4.2.3.0.1 is its conjugacy; id est, one can solve eitherthe primal or dual problem in (649) and then find a solution to the othervia the optimality conditions. When a dual optimal solution is known,for example, a primal optimal solution is any primal feasible solution inhyperplane {X | 〈S ⋆ , X〉=0}.4.14 distinct from maximal complementarity (4.1.2).