Interior-point methods for nonconvex nonlinear programming: regularization and warmstarts

In this paper, we investigate the use of an exact primal-dual penalty approach within the framework of an interior-point method for nonconvex nonlinear programming. This approach provides regularization and relaxation, which can aid in solving ill-behaved problems and in warmstarting the algorithm. We present details of our implementation within the loqo algorithm and provide extensive numerical results on the CUTEr test set and on warmstarting in the context of quadratic, nonlinear, mixed integer nonlinear, and goal programming. Research of the first author is sponsored by ONR grant N00014-04-1-0145. Research of the second author is supported by NSF grant DMS-0107450.     

An exact primal–dual penalty method approach to warmstarting interior-point methods for linear programming

One perceived deficiency of interior-point methods in comparison to active set methods is their inability to efficiently re-optimize by solving closely related problems after a warmstart. In this paper, we investigate the use of a primal–dual penalty approach to overcome this problem. We prove exactness and convergence and show encouraging numerical results on a set of linear and mixed integer programming problems. Research of the first author is sponsored by ONR grant N00014-04-1-0145. Research of the second author is supported by NSF grant DMS-0107450.     

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.     

Robust convex quadratically constrained programs

In this paper we study robust convex quadratically constrained programs, a subset of the class of robust convex programs introduced by Ben-Tal and Nemirovski [4]. In contrast to [4], where it is shown that such robust problems can be formulated as semidefinite programs, our focus in this paper is to identify uncertainty sets that allow this class of problems to be formulated as second-order cone programs (SOCP). We propose three classes of uncertainty sets for which the robust problem can be reformulated as an explicit SOCP and present examples where these classes of uncertainty sets are natural. Research partially supported by DOE grant GE-FG01-92ER-25126, NSF grants DMS-94-14438, CDA-97-26385, DMS-01-04282 and ONR grant N000140310514.Research partially supported by NSF grants CCR-00-09972, DMS-01-04282 and ONR grant N000140310514.     

Combining phase I and phase II in a potential reduction algorithm for linear programming

This paper describes an affine potential reduction algorithm for linear programming that simultaneously seeks feasibility and optimality. The algorithm is closely related to a similar method of Anstreicher. The new features are that we use a two-dimensional programming problem to derive better lower bounds than Anstreicher, that our direction-finding subproblem treats phase I and phase II more symmetrically, and that we do not need an initial lower bound. Our method also allows for the generation of a feasible solution (so that phase I is terminated) during the course of the iterations, and we describe two ways to encourage this behavior.Research supported in part by NSF grant DMS-8904406 and by NSF, AFOSR and ONR through NSF grant DMS-8920550.     

The homology of irregular dihedral branched covers ofS 3

Partially supported by NSF grant DMS-9208052 and the MSRI NSF grant DMS-9022140. The author held an MSRI Research Professorship while the paper was being written     

Scaling,shifting and weighting in interior-point methods

We examine certain questions related to the choice of scaling, shifting and weighting strategies for interior-point methods for linear programming. One theme is the desire to make trajectories to be followed by algorithms into straight lines if possible to encourage fast convergence. While interior-point methods in general follow curves, this occurrence of straight lines seems appropriate to honor George Dantzig's contributions to linear programming, since his simplex method can be seen as following either a piecewise-linear path inn-space or a straight line inm-space (the simplex interpretation).Dedicated to Professor George B. Dantzig on the occasion of his eightieth birthday.Research supported in part by NSF, AFOSR, and ONR through NSF Grant DMS-8920550.     

The homology of cyclic branched covers ofS 3

Partially supported by NSF grant DMS-9208052 and the MSRI NSF grant DMS-9022140. The author held an MSRI Research Professorship while the paper was being written     

A computational study of a gradient-based log-barrier algorithm for a class of large-scale SDPs

 The authors of this paper recently introduced a transformation [4] that converts a class of semidefinite programs (SDPs) into nonlinear optimization problems free of matrix-valued constraints and variables. This transformation enables the application of nonlinear optimization techniques to the solution of certain SDPs that are too large for conventional interior-point methods to handle efficiently. Based on the transformation, we proposed a globally convergent, first-order (i.e., gradient-based) log-barrier algorithm for solving a class of linear SDPs. In this paper, we discuss an efficient implementation of the proposed algorithm and report computational results on semidefinite relaxations of three types of combinatorial optimization problems. Our results demonstrate that the proposed algorithm is indeed capable of solving large-scale SDPs and is particularly effective for problems with a large number of constraints. Received: June 22, 2001 / Accepted: January 20, 2002 Published online: December 9, 2002 RID="†" ID="†"Computational results reported in this paper were obtained on an SGI Origin2000 computer at Rice University acquired in part with support from NSF Grant DMS-9872009. RID="⋆" ID="⋆"This author was supported in part by NSF Grants CCR-9902010, INT-9910084 and CCR-0203426 RID="⋆⋆" ID="⋆⋆"This author was supported in part by NSF Grants CCR-9902010, INT-9910084 and CCR-0203113 RID="⋆⋆⋆" ID="⋆⋆⋆"This author was supported in part by DOE Grant DE-FG03-97ER25331, DOE/LANL Contract 03891-99-23 and NSF Grant DMS-9973339. Key Words. semidefinite program – semidefinite relaxation – nonlinear programming – interior-point methods – limited memory quasi-Newton methods. Mathematics Subject Classification (1991): 90C06, 90C27, 90C30.     

Potential-reduction methods in mathematical programming

We provide a survey of interior-point methods for linear programming and its extensions that are based on reducing a suitable potential function at each iteration. We give a fairly complete overview of potential-reduction methods for linear programming, focusing on the possibility of taking long steps and the properties of the barrier function that are necessary for the analysis. We then describe briefly how the methods and results can be extended to certain convex programming problems, following the approach of Nesterov and Todd. We conclude with some open problems. Research supported in part by NSF, AFOSR and ONR through NSF Grant DMS-8920550. Some of this work was done while the author was on a sabbatical leave from Cornell University visiting the Department of Mathematics at the University of Washington.     

Initial segments of the degrees of constructibility

We show that any constructible, constructibly countable, (dual) algebraic lattice is isomorphic to the degrees of constructibility of reals in some generic extension ofL. Research partially supported by NSF grant DMS-8601777. Research partially supported by NSF grant DMS-8601048 and grant 84-00067 from the U.S.-Israel Binational Science Foundation. Thanks are also due to Uri Abraham and Mati Rubin for helpful discussions about initial segments of the degrees of constructibility and to Bill Lampe for information on algebraic lattices.     

Mixed hyperbolic-elliptic systems in self-similar flows

From the observation that self-similar solutions of conservation laws in two space dimensions change type, it follows that for systems of more than two equations, such as the equations of gas dynamics, the reduced systems will be of mixed hyperbolic-elliptic type, in some regions of space. In this paper, we derive mixed systems for the isentropic and adiabatic equations of compressible gas dynamics. We show that the mixed systems which arise exhibit complicated nonlinear dependence. In a prototype system, the nonlinear wave system, this behavior is much simplified, and we outline the solution to some typical Riemann problems.Dedicated to Constantine Dafermos on his 60^th birthdayResearch supported by the National Science Foundation, grant DMS-9970310.Research supported by the Department of Energy, grant DE-FG-03-94-ER25222 and by the National Science Foundation, grant DMS-9973475 (POWRE).Research supported by the Department of Energy, grant DE-FG-03-94-ER25222 and by the National Science Foundation, grant DMS-0103823.     

The Lévy-Hinčin representation for random compact convex subsets which are infinitely divisible under Minkowski addition

Summary This paper establishes the Lévy-Hinčin representation for all random compact convex subsets of ℝ which are infinitely divisible for Minkowski addition. Research partially supported by NSF grants No. MCS 8100728 and DMS-8318610 Research partially supported by NSF grant No. MCS 8301326     

On polynomiality of the Mehrotra-type predictor—corrector interior-point algorithms

Recently, Mehrotra [3] proposed a predictor—corrector primal—dual interior-point algorithm for linear programming. At each iteration, this algorithm utilizes a combination of three search directions: the predictor, the corrector and the centering directions, and requires only one matrix factorization. At present, Mehrotra's algorithmic framework is widely regarded as the most practically efficient one and has been implemented in the highly successful interior-point code OB1 [2]. In this paper, we study the theoretical convergence properties of Mehrotra's interior-point algorithmic framework. For generality, we carry out our analysis on a horizontal linear complementarity problem that includes linear and quadratic programming, as well as the standard linear complementarity problem. Under the monotonicity assumption, we establish polynomial complexity bounds for two variants of the Mehrotra-type predictor—corrector interior-point algorithms. These results are summarized in the last section in a table.Research supported in part by NSF DMS-9102761, DOE DE-FG05-91ER25100 and DOE DE-FG02-93ER25171.Corresponding author.     

On the convergence of the iteration sequence in primal-dual interior-point methods

Recently, numerous research efforts, most of them concerned with superlinear convergence of the duality gap sequence to zero in the Kojima—Mizuno—Yoshise primal-dual interior-point method for linear programming, have as a primary assumption the convergence of the iteration sequence. Yet, except for the case of nondegeneracy (uniqueness of solution), the convergence of the iteration sequence has been an important open question now for some time. In this work we demonstrate that for general problems, under slightly stronger assumptions than those needed for superlinear convergence of the duality gap sequence (except of course the assumption that the iteration sequence converges), the iteration sequence converges. Hence, we have not only established convergence of the iteration sequence for an important class of problems, but have demonstrated that the assumption that the iteration sequence converges is redundant in many of the above mentioned works.This research was supported in part by NSF Coop. Agr. No. CCR-8809615. A part of this research was performed in June, 1991 while the second and the third authors were at Rice University as visiting members of the Center for Research in Parallel Computation.Corresponding author. Research supported in part by AFOSR 89-0363, DOE DEFG05-86ER25017 and ARO 9DAAL03-90-G-0093.Research supported in part by NSF DMS-9102761 and DOE DE-FG05-91ER25100.Research supported in part by NSF DDM-8922636.     

Gradient estimates,Harnack inequalities and estimates for heat kernels of the sum of squares of vector fields

Research partially supported by NSF grant no. DMS-87-04209 and DMS-90-04062     

On the formulation and theory of the Newton interior-point method for nonlinear programming

In this work, we first study in detail the formulation of the primal-dual interior-point method for linear programming. We show that, contrary to popular belief, it cannot be viewed as a damped Newton method applied to the Karush-Kuhn-Tucker conditions for the logarithmic barrier function problem. Next, we extend the formulation to general nonlinear programming, and then validate this extension by demonstrating that this algorithm can be implemented so that it is locally and Q-quadratically convergent under only the standard Newton method assumptions. We also establish a global convergence theory for this algorithm and include promising numerical experimentation.The first two authors were supported in part by NSF Cooperative Agreement No. CCR-8809615, by Grants AFOSR 89-0363, DOE DEFG05-86ER25017, ARO 9DAAL03-90-G-0093, and the REDI Foundation. The fourth author was supported in part by NSF DMS-9102761 and DOE DE-FG02-93ER25171. The authors would like to thank Sandra Santos for painstakingly proofreading an earlier verion of this paper.     

On multilevel iterative methods for optimization problems

This paper is concerned with multilevel iterative methods which combine a descent scheme with a hierarchy of auxiliary problems in lower dimensional subspaces. The construction of auxiliary problems as well as applications to elasto-plastic model and linear programming are described. The auxiliary problem for the dual of a perturbed linear program is interpreted as a dual of perturbed aggregated linear program. Coercivity of the objective function over the feasible set is sufficient for the boundedness of the iterates. Equivalents of this condition are presented in special cases.Supported by NSF under grant DMS-8704169, AFOSR under grant 86-0126, and ONR under Contract N00014-83-K-0104. Consulting for American Airlines Decision Technologies, MD 2C55, P.O. Box 619616, DFW, TX 75261-9616, USA.Supported by NSF under grant DMS-8704169 and AFOSR under grant 86-0126.     

Restarted block Lanczos bidiagonalization methods

The problem of computing a few of the largest or smallest singular values and associated singular vectors of a large matrix arises in many applications. This paper describes restarted block Lanczos bidiagonalization methods based on augmentation of Ritz vectors or harmonic Ritz vectors by block Krylov subspaces. Research supported in part by NSF grant DMS-0107858, NSF grant DMS-0311786, and an OBR Research Challenge Grant.     

A reduced Hessian method for constrained optimization

Reduced Hessian methods have been shown to be successful for equality constrained problems. However there are few results on reduced Hessian methods for general constrained problems. In this paper we propose a method for general constrained problems, based on Byrd and Schnabel's basis-independent algorithm. It can be regarded as a smooth extension of the standard reduced Hessian Method.Research supported in part by NSF, AFORS and ONR through NSF grant DMS-8920550.