Linear Semidefinite Programming Problems Research Articles

Logarithmic Barrier Method Via Minorant Function for Linear Semidefinite Programming

Abstract We propose in this study, a new logarithmic barrier approach to solve linear semidefinite programming problem. We are interested in computation of the direction by Newton’s method and of the displacement step using minorant functions instead of line search methods in order to reduce the computation cost. Our new approach is even more beneficial than classical line search methods. This purpose is confirmed by some numerical simulations showing the e˙ectiveness of the algorithm developed in this work, which are presented in the last section of this paper.

A smoothing gradient-based neural network strategy for solving semidefinite programming problems

ABSTRACT Linear semidefinite programming problems have received a lot of attentions because of large variety of applications. This paper deals with a smooth gradient neural network scheme for solving semidefinite programming problems. According to some properties of convex analysis and using a merit function in matrix form, a neural network model is constructed. It is shown that the proposed neural network is asymptotically stable and converges to an exact optimal solution of the semidefinite programming problem. Numerical simulations are given to show that the numerical behaviours are in good agreement with the theoretical results.

Primal–Dual Newton Method with Steepest Descent for the Linear Semidefinite Programming Problem: Iterative Process

Primal–Dual Newton Method with Steepest Descent for the Linear Semidefinite Programming Problem: Iterative Process

Primal–Dual Newton Method with Steepest Descent for the Linear Semidefinite Programming Problem: Newton’s System of Equations

Primal–Dual Newton Method with Steepest Descent for the Linear Semidefinite Programming Problem: Newton’s System of Equations

Linear semidefinite programming problems: regularisation and strong dual formulations

Regularisation consists in reducing a given optimisation problem to an equivalent form where certain regularity conditions, which guarantee the strong duality, are fulfilled. In this paper, for linear problems of semidefinite programming (SDP), we propose a regularisation procedure which is based on the concept of an immobile index set and its properties. This procedure is described in the form of a finite algorithm which converts any linear semidefinite problem to a form that satisfies the Slater condition. Using the properties of the immobile indices and the described regularization procedure, we obtained new dual SDP problems in implicit and explicit forms. It is proven that for the constructed dual problems and the original problem the strong duality property holds true.

ON OPTIMUM DESIGN OF FRAME STRUCTURES

Optimization of frame structures is formulated as a non-convex optimization problem, which is currently solved to local optimality. In this contribution, we investigate four optimization approaches: (i) general non-linear optimization, (ii) optimality criteria method, (iii) non-linear semidefinite programming, and (iv) polynomial optimization. We show that polynomial optimization solves the frame structure optimization to global optimality by building the (moment-sums-of-squares) hierarchy of convex linear semidefinite programming problems, and it also provides guaranteed lower and upper bounds on optimal design. Finally, we solve three sample optimization problems and conclude that the local optimization approaches may indeed converge to local optima, without any solution quality measure, or even to infeasible points. These issues are readily overcome by using polynomial optimization, which exhibits a finite convergence, at the prize of higher computational demands.

Two-phase simplex method for linear semidefinite optimization

The linear semidefinite programming problem is considered. To solve it, the variant of the primal simplex method, that generalizes the corresponding method for linear programming problems, is proposed. The passage from one extreme point of the feasible set to another one is described. The main attention is given to pivoting in the case, when the extreme point is irregular, i.e. the “triangular” number of rank of the matrix in the basic point is less than number of equality type constraints in the problem. The approach for finding a starting extreme point is proposed too. The local convergence of the method is proven.

A Variant of the Dual Simplex Method for a Linear Semidefinite Programming Problem

A linear semidefinite programming problem in the standard statement is considered, and a variant of the dual simplex method is proposed for its solution. This variant generalizes the corresponding method used for linear programming problems. The transfer from an extreme point of the feasible set to another extreme point is described. The convergence of the method is proved.

A Sequential Convex Program Approach to an Inverse Linear Semidefinite Programming Problem

This paper is devoted to the study of solving method for a type of inverse linear semidefinite programming problem in which both the objective parameter and the right-hand side parameter of the linear semidefinite programs are required to adjust. Since such kind of inverse problem is equivalent to a mathematical program with semidefinite cone complementarity constraints which is a rather difficult problem, we reformulate it as a nonconvex semi-definte programming problem by introducing a nonsmooth partial penalty function to penalize the complementarity constraint. The penalized problem is actually a nonsmooth DC programming problem which can be solved by a sequential convex program approach. Convergence analysis of the penalty models and the sequential convex program approach are shown. Numerical results are reported to demonstrate the efficiency of our approach.

A feasible dual affine scaling steepest descent method for the linear semidefinite programming problem

The linear semidefinite programming problem is considered. The dual affine scaling method in which all current iterations belong to the feasible set is proposed for its solution. Moreover, the boundaries of the feasible set may be reached. This method is a generalization of a version of the affine scaling method that was earlier developed for linear programs to the case of semidefinite programming.

A feasible primal–dual interior point method for linear semidefinite programming

In this paper, we consider a feasible primal–dual interior point method for linear semidefinite programming problem (SDP) based on Alizadeh–Haeberly–Overton (AHO) direction (Monteiro, 1997). Firstly, and by a new and simple technique, we establish the existence and uniqueness of optimal solution of the perturbed problem (SDP)μ and its convergence to optimal solution of (SDP). Next, we present new different alternatives to calculate the displacement step. After, we establish the convergence of the obtained algorithm and we show that its complexity is O(nln[ε−1(〈X0,S0〉)]). Finally, we present some numerical simulations which show the effectiveness of the algorithm developed in this work.

An admissible dual internal point method for a linear semidefinite programming problem

We consider a linear semidefinite programming problem. To solve this problem, we propose a dual numerical method from the class of affine scaling methods. We use Ostrovskii's theorem to establish its local convergence. We also show results for test problems.

Dual interior point methods for linear semidefinite programming problems

Dual interior point methods for solving linear semidefinite programming problems are proposed. These methods are an extension of dual barrier-projection methods for linear programs. It is shown that the proposed methods converge locally at a linear rate provided that the solutions to the primal and dual problems are nondegenerate.

A Two-Stage Method for the Design of Near-Field Broadband Beamformer

In this paper, the design of near-field broadband beamformer is considered. The idea is to design the beamformer filter coefficients such that the error between the actual response and the desired response is minimized. This problem can be formulated as a minimax optimization problem and can be transformed into a linear semi-definite programming problem. Interior point algorithm can then be applied after discretization. However, if the dimension of the problem is high, the number of discrete points of the specified region and consequently the dimension of the resultant constraint matrix is high. This high dimension provides a difficulty when directly employing interior point algorithm. Since it results in a large size problem, this problem is considered in this paper. To reduce the computational complexity and memory usage, a two-stage method has been proposed. In the first stage, by using an infinite length filter formulation, an infinite length limit for the original problem is obtained. Then, based on that limit, a reduced problem with a finite filter length is found. The computational complexity of this method is derived. It provides a significant reduction compared to a direct solution of the original problem. It is shown that as the filter length increases, the approximative problem comes close to the optimal infinite solution. Furthermore, we demonstrate with examples the close correspondence between proposed design and optimal design. Also the superiority of the suggested method is illustrated in a three-dimensional beamformer design, where the direct method can only design broadband beamformer of very short filter lengths.

Image Space Analysis for Vector Variational Inequalities with Matrix Inequality Constraints and Applications

In this paper, vector variational inequalities (VVI) with matrix inequality constraints are investigated by using the image space analysis. Linear separation for VVI with matrix inequality constraints is characterized by using the saddle-point conditions of the Lagrangian function. Lagrangian-type necessary and sufficient optimality conditions for VVI with matrix inequality constraints are derived by utilizing the separation theorem. Gap functions for VVI with matrix inequality constraints and weak sharp minimum property for the solutions set of VVI with matrix inequality constraints are also considered. The results obtained above are applied to investigate the Lagrangian-type necessary and sufficient optimality conditions for vector linear semidefinite programming problems as well as VVI with convex quadratic inequality constraints.

Regularization Methods for Semidefinite Programming

We introduce a new class of algorithms for solving linear semidefinite programming (SDP) problems. Our approach is based on classical tools from convex optimization such as quadratic regularization and augmented Lagrangian techniques. We study the theoretical properties and we show that practical implementations behave very well on some instances of SDP having a large number of constraints. We also show that the “boundary point method” from Povh, Rendl, and Wiegele [Computing, 78 (2006), pp. 277-286] is an instance of this class.

A primal interior point method for the linear semidefinite programming problem

The linear semidefinite programming problem is examined. A primal interior point method is proposed to solve this problem. It extends the barrier-projection method used for linear programs. The basic properties of the proposed method are discussed, and its local convergence is proved.

Convergence of a Dual-Variable Vector Sequence in a Semi-Definite Programming Problem

An iterative algorithm is proposed in the paper for determination of a feasible solution to a linear semi-definite programming problem. A basic result of convergence of the interior point method is proved. Two strategies of choosing a step value are considered.

Solving a class of semidefinite programs via nonlinear programming

In this paper, we introduce a transformation that converts a class of linear and nonlinear semidefinite programming (SDP) problems into nonlinear optimization problems. For those problems of interest, the transformation replaces matrix-valued constraints by vector-valued ones, hence reducing the number of constraints by an order of magnitude. The class of transformable problems includes instances of SDP relaxations of combinatorial optimization problems with binary variables as well as other important SDP problems. We also derive gradient formulas for the objective function of the resulting nonlinear optimization problem and show that both function and gradient evaluations have affordable complexities that effectively exploit the sparsity of the problem data. This transformation, together with the efficient gradient formulas, enables the solution of very large-scale SDP problems by gradient-based nonlinear optimization techniques. In particular, we propose a first-order log-barrier method designed for solving a class of large-scale linear SDP problems. This algorithm operates entirely within the space of the transformed problem while still maintaining close ties with both the primal and the dual of the original SDP problem. Global convergence of the algorithm is established under mild and reasonable assumptions.