Lipták and Tunçel in [Lipták L. and Tunçel L. The stable set problem and the lift-and-project ranks of graphs, Mathematical programming B 98 (2003) 319-353] study the minimum number of nodes needed...
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We study stochastic linear-quadratic (LQ) optimal control problems over an infinite time horizon, allowing the cost matrices to be indefinite. We develop a systematic approach based on semidefinite programming (SDP). ...
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We study stochastic linear-quadratic (LQ) optimal control problems over an infinite time horizon, allowing the cost matrices to be indefinite. We develop a systematic approach based on semidefinite programming (SDP). A central issue is the stability of the feedback control;and we show this can be effectively examined through the complementary duality of the SDP. Furthermore, we establish several implication relations among the SDP complementary duality, the ( generalized) Riccati equation, and the optimality of the LQ control problem. Based on these relations, we propose a numerical procedure that provides a thorough treatment of the LQ control problem via primal-dual SDP: it identifies a stabilizing feedback control that is optimal or determines that the problem possesses no optimal solution. For the latter case, we develop an epsilon -approximation scheme that is asymptotically optimal.
After a brief overview of the problem of finding the extremal (minimum or maximum) rank positive semi-definite matrix subject to matrix inequalities, we identify a few new classes of such problems that can be efficien...
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ISBN:
(纸本)0780383354
After a brief overview of the problem of finding the extremal (minimum or maximum) rank positive semi-definite matrix subject to matrix inequalities, we identify a few new classes of such problems that can be efficiently solved. We then proceed to present an algorithm for solving the general class of rank minimization problems.
We study a deterministic linear-quadratic (LQ) control problem over an infinite horizon, without the restriction that the control cost matrix R or the state cost matrix Q be positive-definite. We develop a general app...
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We study a deterministic linear-quadratic (LQ) control problem over an infinite horizon, without the restriction that the control cost matrix R or the state cost matrix Q be positive-definite. We develop a general approach to the problem based on semidefinite programming (SDP) and related duality analysis. We show that the complementary duality condition of the SDP is necessary and sufficient for the existence of an optimal LQ control under a certain stability condition (which is satisfied automatically when Q is positive-definite). When the complementary duality does hold, an optimal state feedback control is constructed explicitly in terms of the solution to the primal SDP.
We introduce a new method of constructing approximation algorithms for combinatorial optimization problems using semidefinite programming. It consists of expressing each combinatorial object in the original problem as...
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We introduce a new method of constructing approximation algorithms for combinatorial optimization problems using semidefinite programming. It consists of expressing each combinatorial object in the original problem as a constellation of vectors in the semidefinite program. When we apply this technique to systems of linear equations mod p with at most two variables in each equation, we can show that the problem is approximable within (1 - kappa (p))p, where kappa (p)> 0 for all p. Using standard techniques we also show that it is NP-hard to approximate the problem within a constant ratio, independent of p. (C) 2001 Academic Press.
In this paper we study the properties of the analytic central path of a semidefinite programming problem under perturbation of the right hand side of the constraints, including the limiting behavior when the central o...
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In this paper we study the properties of the analytic central path of a semidefinite programming problem under perturbation of the right hand side of the constraints, including the limiting behavior when the central optimal solution, namely the analytic center of the optimal set, is approached. Our analysis assumes the primal-dual Slater condition and the strict complementarity condition. Our findings are as follows. First, on the negative side, if we view the central optimal solution as a function of the right hand side of the constraints, then this function is not continuous in general, whereas in the linear programming case this function is known to be Lipschitz continuous. On the positive side, compared with the previous conclusion we obtain a (seemingly) paradoxical result: on the central path any directional derivative with respect to the right hand side of the constraints is bounded, and even converges as the central optimal solution is approached. This phenomenon is possible due to the lack of a uniform bound on the derivatives with respect to the right hand side parameters. All these results are based on the strict complementarity assumption. Concerning this last property we give an example. in that example the yet of right hand side parameters for which the strict complementarity condition holds is neither open nor closed. This is remarkable since a similar set for which the primal-dual Slater condition holds is always open.
A critical disadvantage of primal-dual interior-point methods compared to dual interior-point methods for large scale semidefinite programs (SDPs) has been that the primal positive semidefinite matrix variable becomes...
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A critical disadvantage of primal-dual interior-point methods compared to dual interior-point methods for large scale semidefinite programs (SDPs) has been that the primal positive semidefinite matrix variable becomes fully dense in general even when all data matrices are sparse. Based on some fundamental results about positive semidefinite matrix completion, this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage. Our method is used in two ways. One is a conversion of a sparse SDP having a large scale positive semidefinite matrix variable into an SDP having multiple but smaller positive semidefinite matrix variables to which we can effectively apply any interior-point method for SDPs employing a standard block-diagonal matrix data structure. The other way is an incorporation of our method into primal-dual interior-point methods which we can apply directly to a given SDP. In Part II of this article, we will investigate an implementation of such a primal-dual interior-point method based on positive definite matrix completion, and report some numerical results.
Most of the directions used in practical interior-point methods far semidefinite programming try to follow the approach used in linear programming, i.e., they are defined using the optimality conditions which are modi...
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Most of the directions used in practical interior-point methods far semidefinite programming try to follow the approach used in linear programming, i.e., they are defined using the optimality conditions which are modified with a symmetrization of the perturbed complementarity conditions to allow for application of Newton's method. It is now understood that all the Monteiro-Zhang family, which include, among others, the popular AHO, NT, HKM, Gu, and Toh directions, can be expressed as a scaling of the problem data and of the iterate followed by the solution of the AHO system of equations, followed by the inverse scaling. All these directions therefore share a defining system of equations. The focus of this work is to propose a defining system of equations that is essentially different from the AHO system: the over-determined system obtained from the minimization of a nonlinear equation. The resulting solution is called the Gauss-Newton search direction. We state some of the properties of this system that make it attractive for accurate solutions of semidefinite programs. We also offer some preliminary numerical results that highlight the conditioning of the system and the accuracy of the resulting solutions.
We analyze perturbations of the right-hand side and the cost parameters in linear programming (LP) and semidefinite programming (SDP). We obtain tight bounds on the perturbations that allow interior-point method:, to ...
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We analyze perturbations of the right-hand side and the cost parameters in linear programming (LP) and semidefinite programming (SDP). We obtain tight bounds on the perturbations that allow interior-point method:, to recover feasible and near-optimal solutions in a single interior-point iteration. For the unique. nondegenerate solution case in LP, we show that the bounds obtained using interior-point methods compare nicely with the hounds arising from using the optimal basis. We also present explicit bounds for SDP using the Monteiro-Zhang family of search directions and specialize them to the AHO, H..K..M, and NT directions.
In this paper, a detection strategy based on a semidefinite relaxation of the CDMA maximum-likelihood (ML) problem is investigated. Cutting planes are introduced to strengthen the approximation. The semidefinite progr...
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In this paper, a detection strategy based on a semidefinite relaxation of the CDMA maximum-likelihood (ML) problem is investigated. Cutting planes are introduced to strengthen the approximation. The semidefinite program arising from the relaxation can be solved efficiently using interior point methods. These interior point methods have polynomial computational complexity in the number of users. The simulated bit error rate performance demonstrates that this approach provides a good approximation to the ML performance.
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