H. WOLKOWICZ. Explicit solutions for interval semidefinite linear programs. Linear Algebra Appl., 236:95--104, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....conclude, we use the conjugacy formula to study duality relationships for various convex optimization problems posed over the cone of positive semidefinite, real symmetric matrices. Interest in matrix optimization problems (and duality in particular) has been growing in recent years (for instance [27, 23, 1, 33, 35, 28]) The examples we choose are of recent interest in applications of interior point methods (see for example [1, 15, 21, 2] as well as for variational characterizations of certain quasi Newton updates (see for example [9, 34] 2 Conjugates of induced matrix functions We begin with a technical ....

....In this section we will illustrate how the conjugacy formula derived in Section 2 can be used to study duality properties of optimization problems involving real symmetric matrices. In particular we can study analogues of linear programming over the cone of positive semidefinite matrices (see [27, 23, 1, 33, 35, 2]) penalized versions of such problems (see for example [1, 15, 21, 2] and convex optimization problems leading to well known quasi Newton formulae for minimization algorithms [9, 34] Suppose that X and Y are finite dimensional inner product spaces. For functions F : X ( Gamma1; 1] and G : Y ....

H. Wolkowicz. Explicit solutions for interval semidefinite linear programs. Technical Report CORR 93-29, Department of Combinatorics and Optimization, University of Waterloo, 1993.

....convex case was covered in [14] here we use an independent approach to develop the nonconvex case. Since the seminal paper [5] the study of matrix optimization problems (and in particular eigenvalue optimization) has become extremely prominent. A typical constraint is positive semidefiniteness [7, 22, 25, 26], and with the modern trend towards interior point methods, it has become popular to incorporate this constraint by a barrier penalty function (involving the eigenvalues) 16, 1, 12] A related objective function is used in [8] to give an elegant variational characterization of certain ....

.... trend towards interior point methods, it has become popular to incorporate this constraint by a barrier penalty function (involving the eigenvalues) 16, 1, 12] A related objective function is used in [8] to give an elegant variational characterization of certain quasi Newton formulae (see also [25]) One very common objective function is the maximum eigenvalue [17, 18, 21, 12] or more generally, sums of the largest eigenvalues [19, 10] A key step in algorithm development is the investigation of sensitivity results, and hence differentiability questions about the eigenvalues. The standard ....

H. Wolkowicz. Explicit solutions for interval semidefinite linear programs. Technical Report CORR 93-29, Department of Combinatorics and Optimization, University of Waterloo, 1993.

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H. WOLKOWICZ. Explicit solutions for interval semidefinite linear programs. Linear Algebra Appl., 236:95--104, 1996.

....programming, the interior point methods we present do not seem to bothered by degeneracy. Problem (1. 1) can be rephrased using positive semidefinite constraints which avoid the nondifferentiability, see e.g. 12, 1] Interior point methods for problems involving matrix inequalities are studied in [1, 8, 14, 19, 21, 17]. See the latter for a historical overview. In this paper we study two equivalent differentiable formulations to (MMP) The first is derived using duality theory for an equivalent max min trust region subproblem in conjunction with an interior point approach. We then compare this with the second ....

H. WOLKOWICZ. Explicit solutions for interval semidefinite linear programming. Research Report CORR 93-29, University of Waterloo, 1993.

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