Showing papers on "Semidefinite programming published in 1999"

Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones

Abstract: SeDuMi is an add-on for MATLAB, which lets you solve optimization problems with linear, quadratic and semidefiniteness constraints. It is possible to have complex valued data and variables in SeDuMi. Moreover, large scale optimization problems are solved efficiently, by exploiting sparsity. This paper describes how to work with this toolbox.

SDPT3 — A Matlab software package for semidefinite programming, Version 1.3

Abstract: This software package is a MATLAB implementation of infeasible path-following algorithms for solving standard semidefinite programs (SDP). Mehrotra-type predictor-corrector variants are included. Analogous algorithms for the homogeneous formulation of the standard SDP are also implemented. Four types of search directions are available, namely, the AHO, HKM, NT, and GT directions. A few classes of SDP problems are included as well. Numerical results for these classes show that our algorithms are fairly efficient and robust on problems with dimensions of the order of a hundred.

CSDP, A C library for semidefinite programming

Abstract: This paper describes CSDP, a library of routines that implements a predictor corrector variant of the semidefinite programming algorithm of Helmberg, Rendl, Vanderbei, and Wolkowicz. The main advantages of this code are that it can be used as a stand alone solver or as a callable subroutine, that it is written in C for efficiency, that it makes effective use of sparsity in the constraint matrices, and that it includes support for linear inequality constraints in addition to linear equality constraints. We discuss the algorithm used, its computational complexity, and storage requirements. Finally, we present benchmark results for a collection of test problems.

Handbook of Test Problems in Local and Global Optimization

Abstract: Preface. 1. Introduction. 2. Quadratic Programming Problems. 3. Quadratically Constrained Problems. 4. Univariate Polynomial Problems. 5. Bilinear Problems. 6. Biconvex and (D.C.) Problems. 7. Generalized Geometric Programming. 8. Twice Continuously Differentiable NLPs. 9. Bilevel Programming Problems. 10. Complementarity Problems. 11. Semidefinite Programming Problems. 12. Mixed-Integer Nonlinear Problems. 13. Combinatorial Optimization Problems. 14. Nonlinear Systems of Equations. 15. Dynamic Optimization Problems.

A Spectral Bundle Method for Semidefinite Programming

Abstract: A central drawback of primal-dual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices. This restricts applicability to problems of small dimension. Typically, semidefinite relaxations arising in combinatorial applications have sparse and well-structured cost and coefficient matrices of huge order. We present a method that allows us to compute acceptable approximations to the optimal solution of large problems within reasonable time. Semidefinite programming problems with constant trace on the primal feasible set are equivalent to eigenvalue optimization problems. These are convex nonsmooth programming problems and can be solved by bundle methods. We propose replacing the traditional polyhedral cutting plane model constructed from subgradient information by a semidefinite model that is tailored to eigenvalue problems. Convergence follows from the traditional approach but a proof is included for completeness. We present numerical examples demonstrating the efficiency of the approach on combinatorial examples.

A path-following method for solving BMI problems in control

Abstract: We present a path-following (homotopy) method for (locally) solving bilinear matrix inequality (BMI) problems in control. The method is to linearize the BMI using a first order perturbation approximation, and then iteratively compute a perturbation that "slightly" improves the controller performance by solving a semidefinite program. This process is repeated until the desired performance is achieved, or the performance cannot be improved any further. While this is an approximate method for solving BMIs, we present several examples that illustrate the effectiveness of the approach.

Solving Large-Scale Sparse Semidefinite Programs for Combinatorial Optimization

Abstract: We present a dual-scaling interior-point algorithm and show how it exploits the structure and sparsity of some large-scale problems. We solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints. We report the first computational results of interior-point algorithms for approximating maximum cut semidefinite programs with dimension up to 3,000.

Solving Euclidean Distance Matrix Completion Problems Via Semidefinite Programming

Abstract: Given a partial symmetric matrix A with only certain elements specified, the Euclidean distance matrix completion problem (EDMCP) is to find the unspecified elements of A that make A a Euclidean distance matrix (EDM). In this paper, we follow the successful approach in [20] and solve the EDMCP by generalizing the completion problem to allow for approximate completions. In particular, we introduce a primal-dual interior-point algorithm that solves an equivalent (quadratic objective function) semidefinite programming problem (SDP). Numerical results are included which illustrate the efficiency and robustness of our approach. Our randomly generated problems consistently resulted in low dimensional solutions when no completion existed.

Second Order Optimality Conditions Based on Parabolic Second Order Tangent Sets

Abstract: In this paper we discuss second order optimality conditions in optimization problems subject to abstract constraints. Our analysis is based on various concepts of second order tangent sets and parametric duality. We introduce a condition, called second order regularity, under which there is no gap between the corresponding second order necessary and second order sufficient conditions. We show that the second order regularity condition always holds in the case of semidefinite programming.

Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems

Abstract: We present a tool, outward rotations, for enhancing the performance of several semidefinite programming based approximation algorithms. Using outward rotations, we obtain an approximation algorithm for MAX CUT that, in many interesting cases, performs better than the algorithm of Goemans and Williamson. We also obtain an improved approximation algorithm for MAX NAE{3}-SAT. Finally, we provide some evidence that outward rotations can also be used to obtain improved approximation algorithms for MAX NAE-SAT and MAX SAT.

A study of search directions in primal-dual interior-point methods for semidefinite programming

Abstract: We discuss several different search directions which can be used in primal-dual interior-point methods for semidefinite programming problems and investigate their theoretical properties, including scale invariance, primal-dual symmetry, and whether they always generate well-defined directions. Among the directions satisfying all but at most two of these desirable properties are the Alizadeh-Haeberly-Overton, Helmberg-Rendl-Vanderbei-Wolkowicz/Kojima-Shindoh-HaralMonteiro, Nesterov-Todd, Gu, and Toh directions, as well as directions we will call the MTW and Half directions. The first five of these appear to be the best in our limited computational testing also.

SDPLIB 1.2, a library of semidefinite programming test problems

Abstract: SDPLIB is a collection of semidefinite programming (SDP) test problems. The problems are drawn from a variety of applications, including truss topology design, control systems engineering, and relaxations of combinatorial optimization problems. The current version of the library contains a total of 92 SDP problems encoded in a standard format. It is hoped that SDPLIB will stimulate the development of improved software for the solution of SDP problems.

Derandomizing Approximation Algorithms Based on Semidefinite Programming

Abstract: Remarkable breakthroughs have been made recently in obtaining approximate solutions to some fundamental NP-hard problems, namely Max-Cut, Max k-Cut, Max-Sat, Max-Dicut, Max-bisection, k-vertex coloring, maximum independent set, etc. All these breakthroughs involve polynomial time randomized algorithms based upon semidefinite programming, a technique pioneered by Goemans and Williamson. In this paper, we give techniques to derandomize the above class of randomized algorithms, thus obtaining polynomial time deterministic algorithms with the same approximation ratios for the above problems. At the heart of our technique is the use of spherical symmetry to convert a nested sequence of n integrations, which cannot be approximated sufficiently well in polynomial time, to a nested sequence of just a constant number of integrations, which can be approximated sufficiently well in polynomial time.

Semidefinite Relaxations and Lagrangian Duality with Application to Combinatorial Optimization

Abstract: We show that it is fruitful to dualize the integrality constraints in a combinatorial optimization problem. First, this reproduces the known SDP relaxations of the max-cut and max-stable problems. Then we apply the approach to general combinatorial problems. We show that the resulting duality gap is smaller than with the classical Lagrangian relaxation; we also show that linear constraints need a special treatment.

Cones of Matrices and Successive Convex Relaxations of Nonconvex Sets

Abstract: Let F be a compact subset of the n-dimensional Euclidean space Rn represented by (finitely or infinitely many) quadratic inequalities. We propose two methods, one based on successive semidefinite programming (SDP) relaxations and the other on successive linear programming (LP) relaxations. Each of our methods generates a sequence of compact convex subsets Ck (k = 1, 2, . . .) of Rn such that (a) the convex hull of $F \subseteq C_{k+1} \subseteq C_k$ (monotonicity), (b) $\cap_{k=1}^{\infty} C_k = \text{the convex hull of F (asymptotic convergence). Our methods are extensions of the corresponding Lovasz--Schrijver lift-and-project procedures with the use of SDP or LP relaxation applied to general quadratic optimization problems (QOPs) with infinitely many quadratic inequality constraints. Utilizing descriptions of sets based on cones of matrices and their duals, we establish the exact equivalence of the SDP relaxation and the semi-infinite convex QOP relaxation proposed originally by Fujie and Kojima. Using this equivalence, we investigate some fundamental features of the two methods including (a) and (b) above.

Constrained receding horizon predictive control for systems with disturbances

Abstract: A receding horizon predictive control method for systems with input constraints and disturbances is proposed. A polyhedral feasible set of states which is invariant with respect to a given state feedback control law is derived in the presence of bounded disturbances. The proposed predicted control algorithm deploys a strategy in which the current state is steered into the polyhedral invariant feasible set within a finite number N of feasible control moves, despite the presence of disturbances. The future control moves over the horizon N are represented as the sum of the state feedback control and a perturbation; the perturbation term provides the degrees of freedom with which to enlarge the stabilizable set of initial states. The control algorithm is formulated in linear matrix inequalities so that it can be solved using semidefinite programming.

Advanced gain-scheduling techniques for uncertain systems

Abstract: This paper is concerned with the design of gain-scheduled controllers for uncertain linear parameter-varying systems. Two alternative design techniques for constructing such controllers are discussed. Both techniques are amenable to linear matrix inequality problems via a gridding of the parameter space and a selection of basis functions. These problems are then readily solvable using available tools in convex semidefinite programming. When used together, these techniques provide complementary advantages of reduced computational burden and ease of controller implementation. The problem of synthesis for robust performance is then addressed by a new scaling approach for gain-scheduled control. The validity of the theoretical results are demonstrated through a two-link flexible manipulator design example. This is a challenging problem that requires scheduling of the controller in the manipulator geometry and robustness in face of uncertainty in the high-frequency range.

Robust Multiloop PID Controller Design: A Successive Semidefinite Programming Approach

Abstract: The problem of robust multiloop proportional-integral-derivative (PID) controller tuning for multivariable processes is addressed in this paper. The problem is formulated in the ℋ∞ control framewor...

First-order optimality conditions in generalized semi-infinite programming

Abstract: In this paper, we consider a generalized semi-infinite optimization problem where the index set of the corresponding inequality constraints depends on the decision variables and the involved functions are assumed to be continuously differentiable. We derive first-order necessary optimality conditions for such problems by using bounds for the upper and lower directional derivatives of the corresponding optimal value function. In the case where the optimal value function is directly differentiable, we present first-order conditions based on the linearization of the given problem. Finally, we investigate necessary and sufficient first-order conditions by using the calculus of quasidifferentiable functions.

Free Material Design via Semidefinite Programming: The Multiload Case with Contact Conditions

Abstract: Free material design deals with the question of finding the stiffest structure with respect to one or more given loads which can be made when both the distribution of material and the material itself can freely vary. The case of one single load has been discussed in several recent papers, and an efficient numerical approach was presented in [M. Kocvara, M. Zibulevsky, and J. Zow, RAIRO Model. Math. Anal. Numer. 32 (1998), pp. 255--281]. We attack here the multiload situation (understood in the worst-case sense), which is of much more interest for applications but also significantly more challenging from both the theoretical and the numerical points of view. After a series of transformation steps we reach a problem formulation for which we can prove existence of a solution; a suitable discretization leads to a semidefinite programming problem for which modern polynomial time algorithms of interior point type are available. A number of numerical examples demonstrates the efficiency of our approach.

On a Representation of the Matching Polytope Via Semidefinite Liftings

Abstract: We consider the relaxation of the matching polytope defined by the non-negativity and degree constraints. We prove that given an undirected graph on n nodes and the corresponding relaxation of the matching polytope, ⌊n/2⌋ iterations of the Lovasz-Schrijver semidefinite lifting procedure are needed to obtain the matching polytope, in the worst case. We show that ⌊n/2⌋ iterations of the procedure always suffice.

How Good is the Goemans--Williamson MAX CUT Algorithm?

Abstract: The celebrated semidefinite programming algorithm for MAX CUT introduced by Goemans and Williamson was known to have a performance ratio of at least $\alpha=\frac 2 {\pi} \min_{0<\theta\le \pi} \frac \theta {1-\cos \theta}$ ($0.87856<\alpha<0.87857$); the exact performance ratio was unknown. We prove that the performance ratio of their algorithm is exactly $\alpha$. Furthermore, we show that it is impossible to add valid linear constraints to improve the performance ratio.

A Predictor-Corrector Interior-Point Algorithm for the Semidefinite Linear Complementarity Problem Using the Alizadeh--Haeberly--Overton Search Direction

Abstract: This paper proposes a globally convergent predictor-corrector infeasible-interior-point algorithm for the monotone semidefinite linear complementarity problem using the Alizadeh--Haeberly--Overton search direction, and shows its quadratic local convergence under the strict complementarity condition.

Polynomial Convergence of a New Family of Primal-Dual Algorithms for Semidefinite Programming

Abstract: This paper establishes the polynomial convergence of a new class of primal-dual interior-point path-following feasible algorithms for semidefinite programming (SDP) whose search directions are obtained by applying Newton's method to the symmetric central path equation $$ (P X P^T)^{1/2}(P^{-T} S P^{-1}) ( P X P^T)^{1/2} - \mu I =0, $$ where P is a nonsingular matrix. Specifically, we show that the short-step path-following algorithm based on the Frobenius norm neighborhood and the semilong-step path-following algorithm based on the operator 2-norm neighborhood have $O(\sqrt{n}L)$ and O(nL) iteration-complexity bounds, respectively. When P = I, this yields the first polynomially convergent semilong-step algorithm based on a pure Newton direction. Restricting the scaling matrix P at each iteration to a certain subset of nonsingular matrices, we are able to establish an O(n3/2L) iteration complexity for the long-step path-following method. The resulting subclass of search directions contains both the Nesterov--Todd direction and the Helmberg--Rendl--Vanderbei--Wolkowicz/Kojima--Shindoh--Hara/Monteiro direction.