H. WOLKOWICZ. Some applications of optimization in matrix theory. Linear Algebra and its Applications, 40:101--118, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....consider strict matrix inequalities F 0 x 1 F 1 xmFm 0; 2) where the inequality means the lefthand side is a positive de nite matrix. 2.2 Theorems of alternatives Theorems of alternatives give necessary and sucient conditions for solvability of an LMI. We consider two variations [5, 8, 14, 23]. Theorem 1 The strict LMI (2) is feasible if and only there does not exist a Z = Z T 2 R n n that satis es 0 6= Z 0; TrF 0 Z 0; TrF i Z = 0; i = 1; m: 3) The conditions (2) and (3) are called alternatives. The theorem states that exactly one of both alternatives is feasible. A ....

H. Wolkowicz. Some applications of optimization in matrix theory. Linear Algebra and Applications, 40:101-118, 1981.

....(i.e. inequalities with respect to nonpolyhedral convex cones) and linear matrix inequalities in particular. For our purposes the following three theorems will be sufficient. We refer to [BI69, BBI71, CK77, BW81] for more background on theorems of alternative for nonpolyhedral cones, and to [Wol81, Las95, Las97] for results on linear matrix inequalities. 3 Theorem 1 (ALT 1) Exactly one of the following statements is true. 1. There exists an x 2 V with A(x) A 0 0. 2. There exists a Z 2 S with Z 0, A adj (Z) 0, and hA 0 ; Zi S 0. We refer to Appendix A for a proof of this theorem and the other ....

H. Wolkowicz. Some applications of optimization in matrix theory. Linear Algebra and Appl., 40:101--118, 1981.

....is not polyhedral. What can we do if the semide nite program at hand is not strictly feasible If we know the minimal face of the positive semide nite cone that contains the feasible set, then we can project the problem onto this face as described in Example 4. 1 and obtain a well posed problem [44]. The minimal cone can be constructed explicitly if a point in the relative interior of the feasible set is known. 7 Proposition 4.5 Let X 2 X = fX 0 : AX = bg with eigenvalue decomposition X = P P T , P T P = I and 0 diagonal. Denote by SP = PV P T : V 0 the face of S ....

H. Wolkowicz. Some applications of optimization in matrix theory. Linear Algebra and its Applications, 40:101-118, 1981.

....x 2 S 3 (15) Problem (14) is said to satisfy generalized Slater condition if there exists some y 2 rel int Omega such that Ay 0 b 2 rel int S (where rel int denotes the relative interior) The following duality lemma is a special case of a more general result of [20, Theorem 4.1] Lemma 2. 3 [20] (i) weak duality) If y is a feasible solution of (14) and x is a feasible solution of (15) then b t x c t y. 6 (ii) strong duality) Assume that (14) satisfies generalized Slater condition. If both (14) and (15) have a feasible solution, then min c t y = max b t x: iii) ....

H. Wolkowicz, Some applications of optimization in matrix theory, Linear Algebra and Its Applications 40 (1981) 101--118. 26

....problems where K replaces in the primal problem and K replaces in the dual problem. Duffin in [Duf56] was the first one to study such generalized duality theories. Later Hurwicz [Hur58] Ben Israel, Charnes and Kortanek [BICK69] Borwein and Wolkowicz [BW81b, BW81a] and Wolkowicz [Wol81] among others developed more general formulations of the duality theory. For a comprehensive treatment of generalized duality theory from the point of view of infinite dimensional linear programs, see the text of Anderson and Nash [AN87] and for alternative extensions refer to [BW81b, BW81a] It ....

....A vecX = 0, and X 0 and [Roc70] Theorem 9.1, p. 73 implies that K 1 is closed. Now we state the most common form of Farkas lemma as given in Schrijver s text [Sch86] and as extended to the positive semidefinite cone: 1 Alternative extensions without closedness assumption are treated in [BW81b, BW81a, Wol81] 5 Lemma 3 Extended Farkas lemma: Let b 2 m and A 2 m Thetan 2 be a matrix such that its rows A T i: vecA i where A i are symmetric for i = 1; Delta Delta Delta ; m. Furthermore, let there be an m vector y such that Mat(A T y) 0. Then there exists a symmetric matrix X 0, ....

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H. Wolkowicz. Some applications of optimization in matrix theory. Linear Algebra and its Applications, 40:101--118, 1981.

....arise (e.g. Toeplitz structure) 1. 2 Historical overview An early paper on the theoretical properties of semidefinite programs is Bellman and Fan [BF63] Other references discussing optimality conditions are Craven and Mond [CM81] Shapiro [Sha85] Fletcher [Fle85] Allwright [All88] Wolkowicz [Wol81], and Kojima, Kojima and Hara [KKH94] Many researchers have worked on the problem of minimizing the maximum eigenvalue of a symmetric matrix, which can be cast as a semidefinite program (see x2) See, for instance, Cullum, Donath and Wolfe [CDW75] Goh and Teo [GT88] Panier [Pan89] Allwright ....

....If both conditions hold, the optimal sets X opt and Z opt are nonempty. For a proof, see Nesterov and Nemirovsky [NN94, x4.2] or Rockafellar [Roc70, x30] Theorem 1 is an application of standard duality in convex analysis, so the constraint qualification is not surprising or unusual. Wolkowicz [Wol81], and Ramana [Ram93, Ram95, RG95] have formulated two different approaches to a duality theory for semidefinite programming that does not require strict feasibility. For our present purposes, the standard duality outlined above will be sufficient. Assume the optimal sets are nonempty, i.e. there ....

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H. Wolkowicz. Some applications of optimization in matrix theory. Linear Algebra and Appl., 40:101--118, 1981.

....of d, and hence v(u) is di#erentiable at u 0 . 5 NOTES Lagrangian duality is a well developed concept in mathematical programming. Its origins go back to von Neumann s game theory. In the context of semidefinite programming particular examples of duality schemes were considered, for example, in [1, 19, 25]. Example 2.2, of a linear semidefinite program with a duality gap, is taken from [24] The parametric approach to duality, by applying convex analysis to the parametric problem (2.17) was developed in Rockafellar [17, 18] A proof of the Fenchel Moreau duality theorem can be found in [17] ....

Wolkowicz, H., Some applications of optimization in matrix theory. Linear Algebra and Its Applications 1981; 40:101-118.

....programming over cones or cone LP since the set of positive semidefinite matrices constitutes a convex cone. To some extent, semidefinite programming is very similar to linear programming; see Alizadeh [1] for a comparison. It inherits the very elegant duality theory of cone LP (see Wolkowicz [70] and the exposition by Alizadeh [1] The simplex method can be generalized to semidefinite programs (Pataki [57] Given any ffl 0, semidefinite programs can be solved within an additive error of ffl in polynomial time (ffl is part of the input, so the running time dependence on ffl is ....

H. Wolkowicz. Some applications of optimization in matrix theory. Linear Algebra and Its Applications, 40:101--118, 1981.

....X such that X 0 and B( X) 0; 9 2. Problem (D) is called strictly feasible if there exists a feasible point y and t such that A (y) B ( t) C and t 0. The following theorem characterizes the duality of SDP. For a general theorem for cone LP s and its proof, see e.g. [WOL81]. Theorem 2.1 Let (P ) or (D) be strictly feasible. Then: a) Let X and (y; t) be feasible solutions of (P ) and (D) respectively. Then trace CX a t y b t t. b) If one of the problems is infeasible, then the other is infeasible or unbounded. c) Let both (P ) and (D) be feasible, then ....

H. WOLKOWICZ. Some applications of optimization in matrix theory. Linear Algebra and its Applications, 40:101--118, 1981.

....programming problems where K replaces in the primal problem and K replaces in the dual problem. Duffin in [19] was the first one to study such generalized duality theories. Later Hurwicz [34] Ben Israel, Charnes and Kortanek [9] Borwein and Wolkowicz [11, 12] and Wolkowicz [64] among others developed alternative formulations of the duality theory. For a comprehensive treatment of generalized duality theory from the point of view of infinite dimensional linear programs, see the text of Anderson and Nash [3] and for alternative extensions refer to [11, 12] It is worth ....

.... of lemmas for the component wise inequalities, as given for example in Schrijver s text [60] Related extensions for infinite programs have been studied in [34] and [13] and in the case of matrix variables in [14] In all 1 Alternative extensions without closedness assumption are treated in [11, 12, 64] 8 F. Alizadeh of these extensions we need to assume either some closedness criteria, or the lemma must be modified by using cones other than P (as in [64] for instance. We mention a few more: Lemma 2.4. Let A 2 n 2 Thetam be a matrix whose columns are linearly independent and are of ....

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H. Wolkowicz, Some applications of optimization in matrix theory, Linear Algebra Appl., 40 (1981), pp. 101--118.

....(12) Problem (11) is said to satisfy the generalized Slater condition if there exists some y 2 rel int Omega such that Ay Gamma b 2 rel int S (where rel int denotes the relative interior) The following duality lemma is a special case of a more general result of [20, Theorem 4.1] Lemma 2.1. [20] (i) weak duality) If y is a feasible solution of (11) and x is a feasible solution of (12) then b t x c t y. ii) strong duality) Assume that (11) satisfies the generalized Slater condition. If both (11) and (12) have a feasible solution, then minc t y = maxb t x: iii) ....

H. Wolkowicz, Some applications of optimization in matrix theory, Linear Algebra and Its Applications 40 (1981), pp. 101--118.

....may be considered next natural successors to polyhedra, as one moves beyond linear constraints in optimization theory. 1. 1 Background Historically, semidefinite programming has been studied in more general contexts such as convex and cone programming (see [BCK69] BW81] CDW75] and [Wol81]) See also [Fle85] and [Ove92] Further references can be found in [Ali94] However, the more recent surge of interest in SDP was primarily inspired by the work of [GLS84] see [GLS88] Chapter 9) In this work, the authors associate with every graph G, a convex set denoted by TH(G) and show ....

H. Wolkowicz, Some Applications of Optimization in Matrix Theory, Linear Algebra and its Applications, 40(1981), pp. 101-118.

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H. WOLKOWICZ. Some applications of optimization in matrix theory. Linear Algebra and its Applications, 40:101--118, 1981.

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H. WOLKOWICZ. Some applications of optimization in matrix theory. Linear Algebra Appl., 40:101-- 118, 1981.

....; for if x 0 = Gamma1; then the sign of x changes. Finally consider the bound B 3 in the special case that c = 0. The generalized Slater s constraint qualification holds for this program and so the dual can be found from min y max Y0 trace (QY ) Gamma y t (diag (Y ) Gamma e) see e.g. [27]. This can be rewritten as min y max Y0 trace ( Q Gamma Diag (y) Y ) y t e: The inner maximization yields the value 1 unless Q Gamma Diag (y) 0. This implies that the maximum is attained at Y = 0. Therefore the problem reduces to minimize y t e subject to Q Gamma Diag (y) 0: We ....

Wolkowicz, H., Some applications of optimization in matrix theory. Linear Algebra and its Applications, 40:101--118, 1981.

.... in several different areas, e.g. positive definite completion problems, maximum entropy estimation, and bounds for hard combinatorial problems, see e.g. the survey of Vandenberghe and Boyd [42] Though SDP has been studied in the past, as part of the more general cone programming problem, see e.g. [11, 43], there has been a renewed interest due to the successful applications to discrete optimization [27, 14] and to systems theory [5] In addition, the relaxations are equivalent to the reformulation and linearization technique, see e.g. the survey discussion in [40] which provides further evidence ....

H. WOLKOWICZ. Some applications of optimization in matrix theory. Linear Algebra and its Applications, 40:101--118, 1981.

....the vector in n consisting of the diagonal elements of X. Analogously, for a vector x in n , we let Diag(x) denote the diagonal matrix in M n whose diagonal elements are obtained from x. 2 Duality The general duality theory for problems such as (SDP) has been thoroughly studied, see e.g. [30]. We derive the dual to (SDP) directly using Lagrangian methods. Indeed, let denote the optimal objective value for (SDP) Introducing Lagrange multipliers y 2 k and t 2 m for the equality and inequality constraints, respectively, we see that = max X0 min t0;y tr CX y ....

....Lagrangian. We tacitly assume that both problems have feasible solutions. If a constraint qualification holds, then it can be shown that both problems form a pair of dual problems and strong duality holds; i.e. the minimum attained in (SDP) coincides with the maximum attained in (DSDP) see e.g. [30]. A sufficient condition for strong duality to hold is the existence of strictly feasible interior points for both the primal and the dual problem. Weak duality, the max being less than or equal to the min, holds by construction. In our applications, we will focus mostly on the special case ....

H. WOLKOWICZ. Some applications of optimization in matrix theory. Linear Algebra and its Applications, 40:101--118, 1981.

.... was a major area of study both in finite dimensional and abstract spaces e.g. 20, 8, 13, 14] Observations about LP like duality using a Slater like constraint qualification appear in 1963 by Bellman and Fan, 6] while a strong duality without a constraint qualification is developed in [10, 72] with a regularization algorithm in [11] Semidefinite programming itself appeared in the literature in many places, specifically statistics (covariance matrices, clustering) 51] and control theory (Lyapunov stability, Ricatti equations) e.g. 51, 67, 66] where interior point methods such as ....

.... g Gamma1 (S f Gamma S) The optimality conditions without any constraint qualification become: There exists a Lagrange multiplier vector 2 (S f ) such that f(x) g(x) 8x 2 Omega f : In the special case of linear cone programming, where K;L are closed convex cones, we get [72] the following dual pair for which strong duality holds. The primal problem is: minfcx : Ax K b; x L 0g The minimal cones are denoted K f ; L f ; i.e. F ae L f ; and A(F ) ae K f : The dual program for which strong duality and attainment holds is = maxfby : A y ....

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H. WOLKOWICZ. Some applications of optimization in matrix theory. Linear Algebra and its Applications, 40:101--118, 1981.

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H. Wolkowicz. Some applications of optimization in matrix theory. Linear Algebra and Its Applications, 40:101--118, 1981.

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H. Wolkowicz. Some applications of optimization in matrix theory. Linear Algebra and Its Applications, 40:101--118, 1981. 26

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H. Wolkowicz. Some applications of optimization in matrix theory. Linear Algebra and Appl., 40:101--118, 1981.

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H. WOLKOWICZ. Some applications of optimization in matrix theory. Linear Algebra and ist Applications 40: 101--118, 1981.

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H. Wolkowicz, Some applications of optimization in matrix theory. Linear Algebra and its Applications, 40:101--118, 1981.

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H. Wolkowicz, Some Applications of Optimization in Matrix Theory, Linear Algebra and its Applications, 40(1981), pp. 101-118.

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H. Wolkowicz, Some Applications of Optimization in Matrix Theory, Linear Algebra and its Applications, 40(1981), pp. 101-118.

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