A new second-order corrector interior-point algorithm for semidefinite programming

In this paper, we propose a second-order corrector interior-point algorithm for semidefinite programming (SDP). This algorithm is based on the wide neighborhood. The complexity bound is O(?nL){O(\sqrt{n}L)} for the Nesterov-Todd direction, which coincides with the best known complexity results for SDP. To our best knowledge, this is the first wide neighborhood second-order corrector algorithm with the same complexity as small neighborhood interior-point methods for SDP. Some numerical results are provided as well.     

Alternating direction augmented Lagrangian methods for semidefinite programming

We present an alternating direction dual augmented Lagrangian method for solving semidefinite programming (SDP) problems in standard form. At each iteration, our basic algorithm minimizes the augmented Lagrangian function for the dual SDP problem sequentially, first with respect to the dual variables corresponding to the linear constraints, and then with respect to the dual slack variables, while in each minimization keeping the other variables fixed, and then finally it updates the Lagrange multipliers (i.e., primal variables). Convergence is proved by using a fixed-point argument. For SDPs with inequality constraints and positivity constraints, our algorithm is extended to separately minimize the dual augmented Lagrangian function over four sets of variables. Numerical results for frequency assignment, maximum stable set and binary integer quadratic programming problems demonstrate that our algorithms are robust and very efficient due to their ability or exploit special structures, such as sparsity and constraint orthogonality in these problems.     

On filter-successive linearization methods for nonlinear semidefinite programming

In this paper we present a filter-successive linearization method with trust region for solutions of nonlinear semidefinite programming. Such a method is based on the concept of filter for nonlinear programming introduced by Fletcher and Leyffer in 2002. We describe the new algorithm and prove its global convergence under weaker assumptions. Some numerical results are reported and show that the new method is potentially effcient.     

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.     

Exploiting sparsity in primal-dual interior-point methods for semidefinite programming

The Helmberg-Rendl-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro and Nesterov-Todd search directions have been used in many primal-dual interior-point methods for semidefinite programs. This paper proposes an efficient method for computing the two directions when the semidefinite program to be solved is large scale and sparse.     

Sensitivity analysis in linear programming and semidefinite programming using interior-point methods

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 methods 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 bounds 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. Received: December 1999 / Accepted: January 2001?Published online March 22, 2001     

Solving semidefinite programming problems via alternating direction methods

In this paper, we consider an alternating direction algorithm for the solution of semidefinite programming problems (SDP). The main idea of our algorithm is that we reformulate the complementary conditions in the primal–dual optimality conditions as a projection equation. By using this reformulation, we only need to make one projection and solve a linear system of equation with reduced dimension in each iterate. We prove that the generated sequence converges to the solution of the SDP under weak conditions.     

Global convergence of modified augmented Lagrangian methods for nonlinear semidefinite programming

We investigate in this paper global convergence properties of the augmented Lagrangian method for nonlinear semidefinite programming (NLSDP). Four modified augmented Lagrangian methods for solving NLSDP based on different algorithmic strategies are proposed. Possibly infeasible limit points of the proposed methods are characterized. It is proved that feasible limit points that satisfy the Mangasarian-Fromovitz constraint qualification are KKT points of NLSDP without requiring the boundedness condition of the multipliers. Preliminary numerical results are reported to compare the performance of the modified augmented Lagrangian methods.     

Using a factored dual in augmented Lagrangian methods for semidefinite programming

In the context of augmented Lagrangian approaches for solving semidefinite programming problems, we investigate the possibility of eliminating the positive semidefinite constraint on the dual matrix by employing a factorization. Hints on how to deal with the resulting unconstrained maximization of the augmented Lagrangian are given. We further use the approximate maximum of the augmented Lagrangian with the aim of improving the convergence rate of alternating direction augmented Lagrangian frameworks. Numerical results are reported, showing the benefits of the approach.     

On the convergence of augmented Lagrangian methods for nonlinear semidefinite programming

In this paper, we present new convergence properties of the augmented Lagrangian method for nonlinear semidefinite programs (NSDP). Convergence to the approximately global solutions and optimal values of NSDP is first established for a basic augmented Lagrangian scheme under mild conditions, without requiring the boundedness condition of the multipliers. We then propose four modified augmented Lagrangian methods for NSDP based on different algorithmic strategies. We show that the same convergence of the proposed methods can be ensured under weaker conditions.     

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     

Bounds for codes by semidefinite programming

Delsarte’s method and its extensions allow one to consider the upper bound problem for codes in two-point homogeneous spaces as a linear programming problem with perhaps infinitely many variables, which are the distance distribution. We show that using as variables power sums of distances, this problem can be considered as a finite semidefinite programming problem. This method allows one to improve some linear programming upper bounds. In particular, we obtain new bounds of one-sided kissing numbers.     

Strengthened semidefinite programming bounds for codes

We give a hierarchy of semidefinite upper bounds for the maximum size A(n,d) of a binary code of word length n and minimum distance at least d. At any fixed stage in the hierarchy, the bound can be computed (to an arbitrary precision) in time polynomial in n; this is based on a result of de Klerk et al. (Math Program, 2006) about the regular ∗-representation for matrix ∗-algebras. The Delsarte bound for A(n,d) is the first bound in the hierarchy, and the new bound of Schrijver (IEEE Trans. Inform. Theory 51:2859–2866, 2005) is located between the first and second bounds in the hierarchy. While computing the second bound involves a semidefinite program with O(n ⁷) variables and thus seems out of reach for interesting values of n, Schrijver’s bound can be computed via a semidefinite program of size O(n ³), a result which uses the explicit block-diagonalization of the Terwilliger algebra. We propose two strengthenings of Schrijver’s bound with the same computational complexity. Supported by the Netherlands Organisation for Scientific Research grant NWO 639.032.203.     

Interior proximal methods and central paths for convex second-order cone programming

We make a unified analysis of interior proximal methods of solving convex second-order cone programming problems. These methods use a proximal distance with respect to second-order cones which can be produced with an appropriate closed proper univariate function in three ways. Under some mild conditions, the sequence generated is bounded with each limit point being a solution, and global rates of convergence estimates are obtained in terms of objective values. A class of regularized proximal distances is also constructed which can guarantee the global convergence of the sequence to an optimal solution. These results are illustrated with some examples. In addition, we also study the central paths associated with these distance-like functions, and for the linear SOCP we discuss their relations with the sequence generated by the interior proximal methods. From this, we obtain improved convergence results for the sequence for the interior proximal methods using a proximal distance continuous at the boundary of second-order cones.     

Block-diagonal semidefinite programming hierarchies for 0/1 programming

Lovász and Schrijver, and later Lasserre, proposed hierarchies of semidefinite programming relaxations for 0/1 linear programming problems. We revisit these two constructions and propose two new, block-diagonal hierarchies, which are at least as strong as the Lovász-Schrijver hierarchy, but less costly to compute. We report experimental results for the stable set problem of Paley graphs.     

Strange behaviors of interior-point methods for solving semidefinite programming problems in polynomial optimization

We observe that in a simple one-dimensional polynomial optimization problem (POP), the ??optimal?? values of semidefinite programming (SDP) relaxation problems reported by the standard SDP solvers converge to the optimal value of the POP, while the true optimal values of SDP relaxation problems are strictly and significantly less than that value. Some pieces of circumstantial evidences for the strange behaviors of the SDP solvers are given. This result gives a warning to users of the SDP relaxation method for POPs to be careful in believing the results of the SDP solvers. We also demonstrate how SDPA-GMP, a multiple precision SDP solver developed by one of the authors, can deal with this situation correctly.     

A homotopy method for nonlinear semidefinite programming

In this paper, for solving the nonlinear semidefinite programming problem, a homotopy is constructed by using the parameterized matrix inequality constraint. Existence of a smooth path determined by the homotopy equation, which starts from almost everywhere and converges to a Karush–Kuhn–Tucker point, is proven under mild conditions. A predictor-corrector algorithm is given for numerically tracing the smooth path. Numerical tests with nonlinear semidefinite programming formulations of several control design problems with the data contained in COMPl _e ib are done. Numerical results show that the proposed algorithm is feasible and applicable.     

An ϵ-sensitivity analysis for semidefinite programming

We extend the concept of ϵ-sensitivity analysis developed for linear programming to that for semidefinite programming. First, the notion of ϵ-optimality for a given semidefinite programming problem is defined, and then a generic ϵ-sensitivity analysis for semidefinite programming is introduced. Based on the definitions, we develop an implementation of the generic ϵ-sensitivity analysis under perturbations of either the cost parameters or the right-hand side.     

Complementarity and nondegeneracy in semidefinite programming

Primal and dual nondegeneracy conditions are defined for semidefinite programming. Given the existence of primal and dual solutions, it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution. The converses hold if strict complementarity is assumed. Primal and dual nondegeneracy assumptions do not imply strict complementarity, as they do in LP. The primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutionsX andZ. This is in contrast with LP where nondegeneracy assumptions exactly determine the number of variables which are zero. It is shown that primal and dual nondegeneracy and strict complementarity all hold generically. Numerical experiments suggest probability distributions for the ranks ofX andZ which are consistent with the nondegeneracy conditions. Supported in part by the U.S. National Science Foundation grant CCR-9625955. Supported in part by U.S. National Science Foundation grant CCR-9501941 and the U.S. Office of Naval Research grant N00014-96-1-0704. Supported in part by U.S. National Science Foundation grant CCR-9401119.