We present a nonlinear inverse filtering approach to problems such as power spectrum estimation of stationary time series or deconvolution of a blurred image. The technique is based on eigenvalue optimization and a nu...
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We present a nonlinear inverse filtering approach to problems such as power spectrum estimation of stationary time series or deconvolution of a blurred image. The technique is based on eigenvalue optimization and a numerical treatment may therefore be obtained using primal-dual interior-point methods for semidefinite programming.
An example of an SDP (semidefinite program) exhibits a substantial difficulty in proving the superlinear convergence of a direct extension of the Mizuno-Todd-Ye type predictor-corrector primal-dual interior-point meth...
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An example of an SDP (semidefinite program) exhibits a substantial difficulty in proving the superlinear convergence of a direct extension of the Mizuno-Todd-Ye type predictor-corrector primal-dual interior-point method for LPs (linear programs) to SDPs, and suggests that we need to force the generated sequence to converge to a solution tangentially to the central path (or trajectory). A Mizuno-Todd-Ye type predictor-corrector infeasible-interior-point algorithm incorporating this additional restriction for monotone SDLCPs (semidefinite linear complementarity problems) enjoys superlinear convergence under strict complementarity and nondegeneracy conditions. (C) 1998 The Mathematical programming Society, Inc. Published by Elsevier Science B.V.
We study general geometric techniques for bounding the spectral gap of a reversible Markov chain. We show that the best bound obtainable using these techniques can be computed in polynomial time via semidefinite progr...
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The purpose of this paper is to develop certain geometric results concerning the feasible regions of semidefinite Programs, called here Spectrahedra. We first develop a characterization for the faces of spectrahedra. ...
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The purpose of this paper is to develop certain geometric results concerning the feasible regions of semidefinite Programs, called here Spectrahedra. We first develop a characterization for the faces of spectrahedra. More specifically, given a point x in a spectrahedron, we derive an expression for the minimal face containing x. Among other things, this is shown to yield characterizations for extreme points and extreme rays of spectrahedra. We then introduce the notion of an algebraic polar of a spectrahedron, and present its relation to the usual geometric polar.
This paper studies the semidefinite programming SDP problem, i.e., the optimization problem of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matri...
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This paper studies the semidefinite programming SDP problem, i.e., the optimization problem of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First the classical cone duality is reviewed as it is specialized to SDP is reviewed. Next an interior point algorithm is presented that converges to the optimal solution in polynomial time. The approach is a direct extension of Ye's projective method for linear programming. It is also argued that many known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity carrying over in a similar fashion. Finally, the significance of these results is studied in a variety of combinatorial optimization problems including the general 0-1 integer programs, the maximum clique and maximum stable set problems in perfect graphs, the maximum k-partite subgraph problem in graphs, and various graph partitioning and cut problems. As a result, barrier oracles are presented for certain combinatorial optimization problems (in particular, clique and stable set problem for perfect graphs) whose linear programming formulation requires exponentially many inequalities. Existence of such barrier oracles refutes the commonly believed notion that to solve a combinatorial optimization problem with interior point methods, its linear programming formulation is needed explicity.
Numerous problems in control and systems theory can be formulated in terms of linear matrix inequalities (LMI). Since solving an LMI amounts to a convex optimization problem, such formulations are known to be numerica...
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Numerous problems in control and systems theory can be formulated in terms of linear matrix inequalities (LMI). Since solving an LMI amounts to a convex optimization problem, such formulations are known to be numerically tractable. However, the interest in LMI-based design techniques has really surged with the introduction of efficient interior-point methods for solving LMIs with a polynomial-time complexity, This paper describes one particular method called the Projective Method. Simple geometrical arguments are used to clarify the strategy and convergence mechanism of the Projective algorithm. A complexity analysis is provided, and applications to two generic LMI problems (feasibility and linear objective minimization) are discussed. (C) 1997 The Mathematical programming Society, Inc. Published by Elsevier Science B.V.
Primal-dual pairs of semidefinite programs provide a general framework for the theory and algorithms for the trust region subproblem (TRS), This latter problem consists in minimizing a general quadratic function subje...
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Primal-dual pairs of semidefinite programs provide a general framework for the theory and algorithms for the trust region subproblem (TRS), This latter problem consists in minimizing a general quadratic function subject to a convex quadratic constraint and, therefore, it is a generalization of the minimum eigenvalue problem. The importance of (TRS) is due to the fact that it provides the step in trust region minimization algorithms, The semidefinite framework is studied as an interesting instance of semidefinite programming as well as a tool for viewing known algorithms and deriving new algorithms for (TRS). In particular, a dual simplex type method is studied that solves (TRS) as a parametric eigenvalue problem. This method uses the Lanczos algorithm for the smallest eigenvalue as a black box, Therefore, the essential cost of the algorithm is the matrix-vector multiplication and, thus, sparsity can be exploited. A primal simplex type method provides steps for the so-called hard case, Extensive numerical tests for large sparse problems are discussed. These tests show that the cost of the algorithm is 1 + alpha(n) times the cost of finding a minimum eigenvalue using the Lanczos algorithm, where 0 < alpha(n) < 1 is a fraction which decreases as the dimension increases. (C) 1997 The Mathematical programming Society, Inc. Published by Elsevier Science B.V.
We discuss some consequences of the measure concentration phenomenon for optimization and computational problems. Topics include average case analysis in optimization, efficient approximate counting, computation of mi...
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We discuss some consequences of the measure concentration phenomenon for optimization and computational problems. Topics include average case analysis in optimization, efficient approximate counting, computation of mixed discriminants and permanents, and semidefinite relaxation in quadratic programming. (C) 1997 The Mathematical programming Society, Inc. Published by Elsevier Science B.V.
In this paper we study nonlinear semidefinite programming problems. Convexity, duality and first-order optimality conditions for such problems are presented. A second-order analysis is also given. Second-order necessa...
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In this paper we study nonlinear semidefinite programming problems. Convexity, duality and first-order optimality conditions for such problems are presented. A second-order analysis is also given. Second-order necessary and sufficient optimality conditions are derived. Finally, sensitivity analysis of such programs is discussed, (C) 1997 The Mathematical programming Society, Inc. Published by Elsevier Science B.V.
We consider least-squares problems where the coefficient matrices A, b are unknown but bounded. We minimize the worst-case residual error using (convex) second-order cone programming, yielding an algorithm with comple...
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We consider least-squares problems where the coefficient matrices A, b are unknown but bounded. We minimize the worst-case residual error using (convex) second-order cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of solution and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomial-time using semidefinite programming (SDP). We also consider the case when A, b are rational functions of an unknown-but-bounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worst-case residual. We provide numerical examples, including one from robust identification and one from robust interpolation.
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