We consider a Newton-CG augmented Lagrangian method for solving semidefinite programming (SDP) problems from the perspective of approximate semismooth Newton methods. In order to analyze the rate of convergence of our...
详细信息
We consider a Newton-CG augmented Lagrangian method for solving semidefinite programming (SDP) problems from the perspective of approximate semismooth Newton methods. In order to analyze the rate of convergence of our proposed method, we characterize the Lipschitz continuity of the corresponding solution mapping at the origin. For the inner problems, we show that the positive definiteness of the generalized Hessian of the objective function in these inner problems, a key property for ensuring the efficiency of using an inexact semismooth Newton-CG method to solve the inner problems, is equivalent to the constraint nondegeneracy of the corresponding dual problems. Numerical experiments on a variety of large-scale SDP problems with the matrix dimension n up to 4, 110 and the number of equality constraints m up to 2, 156, 544 show that the proposed method is very efficient. We are also able to solve the SDP problem fap36 (with n = 4, 110 and m = 1, 154, 467) in the Seventh DIMACS Implementation Challenge much more accurately than in previous attempts.
Detecting critical nodes in complex networks (CNP) has great theoretical and practical significance in many disciplines. The existing formulations for CNP are mostly, as we know, based on the integer linear programmin...
详细信息
Detecting critical nodes in complex networks (CNP) has great theoretical and practical significance in many disciplines. The existing formulations for CNP are mostly, as we know, based on the integer linear programming model. However, we observed that, these formulations only considered the sizes but neglected the cohesiveness properties of the connected components in the induced network. To solve the problem and improve the performance of CNP solutions, we construct a novel nonconvex quadratically constrained quadratic programming (QCQP) model and derive its approximation solutions via semidefinite programming (SDP) technique and heuristic algorithms. Various types of synthesized and real-world networks, in the context of different connectivity patterns, are used to validate and verify the effectiveness of the proposed model and algorithm. Experimental results show that our method improved the state of the art of the CNP. (C) 2016 Elsevier B.V. All rights reserved.
Positive semidefinite Hankel matrices arise in many important applications. Some of their properties may be lost due to rounding or truncation errors incurred during evaluation. The problem is to find the nearest matr...
详细信息
Positive semidefinite Hankel matrices arise in many important applications. Some of their properties may be lost due to rounding or truncation errors incurred during evaluation. The problem is to find the nearest matrix to a given matrix to retrieve these properties. The problem is converted into a semidefinite programming problem as well as a problem comprising a semidefined program and second-order cone problem. The duality and optimality conditions are obtained and the primal-dual algorithm is outlined. Explicit expressions for a diagonal preconditioned and crossover criteria have been presented. Computational results are presented. A possibility for further improvement is indicated. (c) 2006 Elsevier B.V. All rights reserved.
We extend the concept of epsilon-sensitivity analysis developed for linear programming to that for semidefinite programming. First, the notion of epsilon-optimality for a given semidefinite programming problem is defi...
详细信息
We extend the concept of epsilon-sensitivity analysis developed for linear programming to that for semidefinite programming. First, the notion of epsilon-optimality for a given semidefinite programming problem is defined, and then a generic c-sensitivity analysis for semidefinite programming is introduced. Based on the definitions, we develop an implementation of the generic epsilon-sensitivity analysis under perturbations of either the cost parameters or the right-hand side. (C) 2004 Elsevier B.V. All rights reserved.
This paper presents a system identification technique for generating stable compact models of typical analog circuit blocks in radio frequency systems. The identification procedure is based on minimizing the model err...
详细信息
This paper presents a system identification technique for generating stable compact models of typical analog circuit blocks in radio frequency systems. The identification procedure is based on minimizing the model error over a given training data set subject to an incremental stability constraint, which is formulated as a semidefinite optimization problem. Numerical results are presented for several analog circuits, including a distributed power amplifier, as well as a MEM device. It is also shown that our dynamical models can accurately predict important circuit performance metrics, and may thus, be useful for design optimization of analog systems.
A wide variety of nonlinear convex optimization problems can be cast as problems involving Linear matrix inequalities (LMIs), and hence efficiently solved using recently developed interior-point methods. In this paper...
详细信息
A wide variety of nonlinear convex optimization problems can be cast as problems involving Linear matrix inequalities (LMIs), and hence efficiently solved using recently developed interior-point methods. In this paper, we will consider two classes of optimization problems with LMI constraints: (1) The semidefinite programming problem, i.e., the problem of minimizing a linear function subject to a linear matrix inequality. semidefinite programming is an important numerical tool for analysis and synthesis in systems and control theory. It has also been recognized in combinatorial optimization as a valuable technique for obtaining bounds on the solution of NP-hard problems. (2) The problem of maximizing the determinant of a positive definite matrix subject to Linear matrix inequalities. This problem has applications in computational geometry, experiment design, information and communication theory, and other fields. We review some of these applications, including some interesting applications that are less well known and arise in statistics, optimal experiment design and VLSI. (C) 1999 Elsevier Science B.V, and IMACS. All rights reserved.
A quadratic surface in n-dimensional space is defined as the locus of zeros of a quadratic polynomial. The quadratic polynomial may be compactly written in notation by an (n + 1)-vector and a real symmetric matrix of ...
详细信息
A quadratic surface in n-dimensional space is defined as the locus of zeros of a quadratic polynomial. The quadratic polynomial may be compactly written in notation by an (n + 1)-vector and a real symmetric matrix of order n + 1, where the vector represents homogenous coordinates of an n-D point, and the symmetric matrix is constructed from the quadratic coefficients. If an n-D quadratic surface is an n-D ellipsoid, the leading n x n principal submatrix of the symmetric matrix would be positive or opposite definite. As we know, to impose a matrix being positive or opposite definite, perhaps the best choice may be to employ semidefinite programming (SDP). From such straightforward and intuitive knowledge, in the literature until 2002, Calafiore first proposed a feasible method for multidimensional ellipsoid-specific fitting using SDP, which minimizes the 2-norm of the algebraic residual vector. However, the runtime of the method is significantly long and memory is often out when the number of fitted points is greater than several thousand. In this paper, we propose a fast and easily implemented algorithm for multidimensional ellipsoid-specific fitting by minimizing a new defined vector norm of the algebraic residual vector using SDP, which drastically decreases the size of the SDP problem while preserving accuracy. The proposed fast method can handle several million fitted points without any difficulty.
The paper considers nonconvex quadratic semidefinite problems. This class arises, for instance, as subproblems in the sequential semidefinite programming algorithm for solving general smooth nonlinear semidefinite pro...
详细信息
The paper considers nonconvex quadratic semidefinite problems. This class arises, for instance, as subproblems in the sequential semidefinite programming algorithm for solving general smooth nonlinear semidefinite problems. We extend locally the concept of self-concordance to problems that satisfy a weak version of the second order sufficient optimality conditions.
A critical obstacle for ultra-wideband (UWB) communications is conformity to restrictions set on the allowed interference to other wireless devices. To this end, UWB signals have to comply with stringent constraints o...
详细信息
A critical obstacle for ultra-wideband (UWB) communications is conformity to restrictions set on the allowed interference to other wireless devices. To this end, UWB signals have to comply with stringent constraints on their emitted power, defined by the Federal Communications Commission spectral mask. Different UWB pulseshaper designs have been studied to meet the spectral mask, out of which an approach based on digital finite impulse response filter design via semidefinite programming has stood out. However, so far this approach has assumed an ideal basic analog pulse to use piece-wise constant constraints for the digital filter design. Since any practical analog pulse does not have a flat spectrum, using piece-wise constant constraints leads to considerable power loss. Avoiding such a loss has motivated us to implement the exact constraints through nonconstant piece-wise continuous bounds. Relative to the design assuming an ideal basic analog pulse, our design examples show that the transmission power can be enhanced considerably while obeying the spectral mask. Such an improvement comes with no extra cost of implementation complexity.
The well known constant rank constraint qualification [Math. Program. Study 21:110-126, 1984] introduced by Janin for nonlinear programming has been recently extended to a conic context by exploiting the eigenvector s...
详细信息
The well known constant rank constraint qualification [Math. Program. Study 21:110-126, 1984] introduced by Janin for nonlinear programming has been recently extended to a conic context by exploiting the eigenvector structure of the problem. In this paper we propose a more general and geometric approach for defining a new extension of this condition to the conic context. The main advantage of our approach is that we are able to recast the strong second-order properties of the constant rank condition in a conic context. In particular, we obtain a second-order necessary optimality condition that is stronger than the classical one obtained under Robinson's constraint qualification, in the sense that it holds for every Lagrange multiplier, even though our condition is independent of Robinson's condition.
暂无评论