The many facets of linear programming

We examine the history of linear programming from computational, geometric, and complexity points of view, looking at simplex, ellipsoid, interior-point, and other methods. Received: June 22, 2000 / Accepted: April 4, 2001?Published online October 2, 2001     

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. Received: January 5, 2000 / Accepted: October 2001?Published online February 14, 2002     

A new bound for the quadratic assignment problem based on convex quadratic programming

We describe a new convex quadratic programming bound for the quadratic assignment problem (QAP). The construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for QAP. The new bound dominates the well-known projected eigenvalue bound, and appears to be competitive with existing bounds in the trade-off between bound quality and computational effort. Received: February 2000 / Accepted: November 2000?Published online January 17, 2001     

An improved rounding method and semidefinite programming relaxation for graph partition

Given an undirected graph G=(V,E) with |V|=n and an integer k between 0 and n, the maximization graph partition (MAX-GP) problem is to determine a subset S⊂V of k nodes such that an objective function w(S) is maximized. The MAX-GP problem can be formulated as a binary quadratic program and it is NP-hard. Semidefinite programming (SDP) relaxations of such quadratic programs have been used to design approximation algorithms with guaranteed performance ratios for various MAX-GP problems. Based on several earlier results, we present an improved rounding method using an SDP relaxation, and establish improved approximation ratios for several MAX-GP problems, including Dense-Subgraph, Max-Cut, Max-Not-Cut, and Max-Vertex-Cover. Received: March 10, 2000 / Accepted: July 13, 2001?Published online February 14, 2002     

Implementation of warm-start strategies in interior-point methods for linear programming in fixed dimension

We implement several warm-start strategies in interior-point methods for linear programming (LP). We study the situation in which both the original LP instance and the perturbed one have exactly the same dimensions. We consider different types of perturbations of data components of the original instance and different sizes of each type of perturbation. We modify the state-of-the-art interior-point solver PCx in our implementation. We evaluate the effectiveness of each warm-start strategy based on the number of iterations and the computation time in comparison with “cold start” on the NETLIB test suite. Our experiments reveal that each of the warm-start strategies leads to a reduction in the number of interior-point iterations especially for smaller perturbations and for perturbations of fewer data components in comparison with cold start. On the other hand, only one of the warm-start strategies exhibits better performance than cold start in terms of computation time. Based on the insight gained from the computational results, we discuss several potential improvements to enhance the performances of such warm-start strategies. This research was supported in part by NSF through CAREER grant DMI-0237415.     

Interior-point methods for linear programming: a review

The paper reviews some recent advances in interior-point methods for linear programming and indicates directions in which future progress can be made. Most of the interior-point methods belong to any of three categories: affine-scaling methods, potential reduction methods and central path methods. These methods are discussed together with infeasible interior methods and homogeneous self-dual methods for linear programming. Also discussed are some theoretical issues in interior-point methods like dependence of complexity bounds on some non-traditional measures different from the input length L of the problem. Finally, the paper concludes with remarks on the comparison of interior-point methods with the simplex method based on their performance on NITLIB suite, a standard collection of test problems.     

A SUCCESSIVE QUADRATIC PROGRAMMING ALGORITHM FOR SDP RELAXATION OF MAX-BISECTION

A successive quadratic programming algorithm for solving SDP relaxation of Max- Bisection is provided and its convergence result is given.The step-size in the algorithm is obtained by solving n easy quadratic equations without using the linear search technique.The numerical experiments show that this algorithm is rather faster than the interior-point method.     

Global optimization of nonconvex factorable programming problems

In this paper, we consider a special class of nonconvex programming problems for which the objective function and constraints are defined in terms of general nonconvex factorable functions. We propose a branch-and-bound approach based on linear programming relaxations generated through various approximation schemes that utilize, for example, the Mean-Value Theorem and Chebyshev interpolation polynomials coordinated with a Reformulation-Linearization Technique (RLT). A suitable partitioning process is proposed that induces convergence to a global optimum. The algorithm has been implemented in C++ and some preliminary computational results are reported on a set of fifteen engineering process control and design test problems from various sources in the literature. The results indicate that the proposed procedure generates tight relaxations, even via the initial node linear program itself. Furthermore, for nine of these fifteen problems, the application of a local search method that is initialized at the LP relaxation solution produced the actual global optimum at the initial node of the enumeration tree. Moreover, for two test cases, the global optimum found improves upon the solutions previously reported in the source literature. Received: January 14, 1998 / Accepted: June 7, 1999?Published online December 15, 2000     

Exploiting special structure in semidefinite programming: A survey of theory and applications

Semidefinite programming (SDP) may be seen as a generalization of linear programming (LP). In particular, one may extend interior point algorithms for LP to SDP, but it has proven much more difficult to exploit structure in the SDP data during computation.     

Logarithmic SUMT limits in convex programming

The limits of a class of primal and dual solution trajectories associated with the Sequential Unconstrained Minimization Technique (SUMT) are investigated for convex programming problems with non-unique optima. Logarithmic barrier terms are assumed. For linear programming problems, such limits – of both primal and dual trajectories – are strongly optimal, strictly complementary, and can be characterized as analytic centers of, loosely speaking, optimality regions. Examples are given, which show that those results do not hold in general for convex programming problems. If the latter are weakly analytic (Bank et al. [3]), primal trajectory limits can be characterized in analogy to the linear programming case and without assuming differentiability. That class of programming problems contains faithfully convex, linear, and convex quadratic programming problems as strict subsets. In the differential case, dual trajectory limits can be characterized similarly, albeit under different conditions, one of which suffices for strict complementarity. Received: November 13, 1997 / Accepted: February 17, 1999?Published online February 22, 2001     

Robust optimization – methodology and applications

Robust Optimization (RO) is a modeling methodology, combined with computational tools, to process optimization problems in which the data are uncertain and is only known to belong to some uncertainty set. The paper surveys the main results of RO as applied to uncertain linear, conic quadratic and semidefinite programming. For these cases, computationally tractable robust counterparts of uncertain problems are explicitly obtained, or good approximations of these counterparts are proposed, making RO a useful tool for real-world applications. We discuss some of these applications, specifically: antenna design, truss topology design and stability analysis/synthesis in uncertain dynamic systems. We also describe a case study of 90 LPs from the NETLIB collection. The study reveals that the feasibility properties of the usual solutions of real world LPs can be severely affected by small perturbations of the data and that the RO methodology can be successfully used to overcome this phenomenon. Received: May 24, 2000 / Accepted: September 12, 2001?Published online February 14, 2002     

Exploiting sparsity in semidefinite programming via matrix completion II: implementation and numerical results

 In Part I of this series of articles, we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs (SDPs) when solving them by primal-dual interior-point methods. This framework is based on some results about positive semidefinite matrix completion, and it can be embodied in two different ways. One is by a conversion of a given sparse SDP having a large scale positive semidefinite matrix variable into an SDP having multiple but smaller positive semidefinite matrix variables. The other is by incorporating a positive definite matrix completion itself in a primal-dual interior-point method. The current article presents the details of their implementations. We introduce new techniques to deal with the sparsity through a clique tree in the former method and through new computational formulae in the latter one. Numerical results over different classes of SDPs show that these methods can be very efficient for some problems. Received: March 18, 2001 / Accepted: May 31, 2001 Published online: October 9, 2002 RID="⋆" ID="⋆"The author was supported by The Ministry of Education, Culture, Sports, Science and Technology of Japan. Key Words. semidefinite programming – primal-dual interior-point method – matrix completion problem – clique tree – numerical results Mathematics Subject Classification (2000): 90C22, 90C51, 05C50, 05C05     

Theory of semidefinite programming for Sensor Network Localization

We analyze the semidefinite programming (SDP) based model and method for the position estimation problem in sensor network localization and other Euclidean distance geometry applications. We use SDP duality and interior-point algorithm theories to prove that the SDP localizes any network or graph that has unique sensor positions to fit given distance measures. Therefore, we show, for the first time, that these networks can be localized in polynomial time. We also give a simple and efficient criterion for checking whether a given instance of the localization problem has a unique realization in using graph rigidity theory. Finally, we introduce a notion called strong localizability and show that the SDP model will identify all strongly localizable sub-networks in the input network. A preliminary version of this paper has appeared in the Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2005.     

Polynomial convergence of primal-dual algorithms for the second-order cone program based on the MZ-family of directions

In this paper we study primal-dual path-following algorithms for the second-order cone programming (SOCP) based on a family of directions that is a natural extension of the Monteiro-Zhang (MZ) family for semidefinite programming. We show that the polynomial iteration-complexity bounds of two well-known algorithms for linear programming, namely the short-step path-following algorithm of Kojima et al. and Monteiro and Adler, and the predictor-corrector algorithm of Mizuno et al., carry over to the context of SOCP, that is they have an O( logε^-1) iteration-complexity to reduce the duality gap by a factor of ε, where n is the number of second-order cones. Since the MZ-type family studied in this paper includes an analogue of the Alizadeh, Haeberly and Overton pure Newton direction, we establish for the first time the polynomial convergence of primal-dual algorithms for SOCP based on this search direction. Received: June 5, 1998 / Accepted: September 8, 1999?Published online April 20, 2000     

Optimality conditions for nonconvex semidefinite programming

This paper concerns nonlinear semidefinite programming problems for which no convexity assumptions can be made. We derive first- and second-order optimality conditions analogous to those for nonlinear programming. Using techniques similar to those used in nonlinear programming, we extend existing theory to cover situations where the constraint matrix is structurally sparse. The discussion covers the case when strict complementarity does not hold. The regularity conditions used are consistent with those of nonlinear programming in the sense that the conventional optimality conditions for nonlinear programming are obtained when the constraint matrix is diagonal. Received: May 15, 1998 / Accepted: April 12, 2000?Published online May 12, 2000     

Interior-point algorithms for semi-infinite programming

In order to study the behavior of interior-point methods on very large-scale linear programming problems, we consider the application of such methods to continuous semi-infinite linear programming problems in both primal and dual form. By considering different discretizations of such problems we are led to a certain invariance property for (finite-dimensional) interior-point methods. We find that while many methods are invariant, several, including all those with the currently best complexity bound, are not. We then devise natural extensions of invariant methods to the semi-infinite case. Our motivation comes from our belief that for a method to work well on large-scale linear programming problems, it should be effective on fine discretizations of a semi-infinite problem and it should have a natural extension to the limiting semi-infinite case.Research supported in part by NSF, AFORS and ONR through NSF grant DMS-8920550.     

Nonlinear programming without a penalty function

In this paper the solution of nonlinear programming problems by a Sequential Quadratic Programming (SQP) trust-region algorithm is considered. The aim of the present work is to promote global convergence without the need to use a penalty function. Instead, a new concept of a “filter” is introduced which allows a step to be accepted if it reduces either the objective function or the constraint violation function. Numerical tests on a wide range of test problems are very encouraging and the new algorithm compares favourably with LANCELOT and an implementation of Sl₁QP. Received: October 17, 1997 / Accepted: August 17, 2000?Published online September 3, 2001     

The Complex Interior-Boundary method for linear and nonlinear programming with linear constraints

In this paper we develop the Complex method; an algorithm for solving linear programming (LP) problems with interior search directions. The Complex Interior-Boundary method (as the name suggests) moves in the interior of the feasible region from one boundary point to another of the feasible region bypassing several extreme points at a time. These directions of movement are guaranteed to improve the objective function. As a result, the Complex method aims to reach the optimal point faster than the Simplex method on large LP programs. The method also extends to nonlinear programming (NLP) with linear constraints as compared to the generalized-reduced gradient.The Complex method is based on a pivoting operation which is computationally efficient operation compared to some interior-point methods. In addition, our algorithm offers more flexibility in choosing the search direction than other pivoting methods (such as reduced gradient methods). The interior direction of movement aims at reducing the number of iterations and running time to obtain the optimal solution of the LP problem compared to the Simplex method. Furthermore, this method is advantageous to Simplex and other convex programs in regard to starting at a Basic Feasible Solution (BFS); i.e. the method has the ability to start at any given feasible solution.Preliminary testing shows that the reduction in the computational effort is promising compared to the Simplex method.     

On sensitivity of central solutions in semidefinite programming

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 set 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. Received: April 2, 1998 / Accepted: January 16, 2001?Published online March 22, 2001     

Interior-point methods for nonconvex nonlinear programming: orderings and higher-order methods

The paper extends prior work by the authors on loqo, an interior point algorithm for nonconvex nonlinear programming. The specific topics covered include primal versus dual orderings and higher order methods, which attempt to use each factorization of the Hessian matrix more than once to improve computational efficiency. Results show that unlike linear and convex quadratic programming, higher order corrections to the central trajectory are not useful for nonconvex nonlinear programming, but that a variant of Mehrotra’s predictor-corrector algorithm can definitely improve performance. Received: May 3, 1999 / Accepted: January 24, 2000?Published online March 15, 2000