Some recent advances in projection-type methods for variational inequalities

AbstractProjection-type methods are a class of simple methods for solving variational inequalities, especially for complementarity problems. In this paper we review and summarize recent developments in this class of methods, and focus mainly on some new trends in projection-type methods

An efficient algorithm for the ℓp\ell_{p} norm based metric nearness problem

Given a dissimilarity matrix, the metric nearness problem is to find the nearest matrix of distances that satisfy the triangle inequalities. This problem has wide applications, such as sensor networks, image processing, and so on. But it is of great challenge even to obtain a moderately accurate solution due to the $O(n^{3})$ metric constraints and the nonsmooth objective function which is usually a weighted $\ell_{p}$ norm based distance. In this paper, we propose a delayed constraint generation method with each subproblem solved by the semismooth Newton based proximal augmented Lagrangian method (PALM) for the metric nearness problem. Due to the high memory requirement for the storage of the matrix related to the metric constraints, we take advantage of the special structure of the matrix and do not need to store the corresponding constraint matrix. A pleasing aspect of our algorithm is that we can solve these problems involving up to $10^{8}$ variables and $10^{13}$ constraints. Numerical experiments demonstrate the efficiency of our algorithm. In theory, firstly, under a mild condition, we establish a primal-dual error bound condition which is very essential for the analysis of local convergence rate of PALM. Secondly, we prove the equivalence between the dual nondegeneracy condition and nonsingularity of the generalized Jacobian for the inner subproblem of PALM. Thirdly, when $q(\cdot)=\|\cdot\|_{1}$ or $\|\cdot\|_{\infty}$, without the strict complementarity condition, we also prove the equivalence between the the dual nondegeneracy condition and the uniqueness of the primal solution

On Polynomial-time Path-following Interior-point Methods with Local Superlinear Convergence

Interior-point methods provide one of the most popular ways of solving convex optimization problems. Two advantages of modern interior-point methods over other approaches are: (1) robust global convergence, and (2) the ability to obtain high accuracy solutions in theory (and in practice, if the algorithms are properly implemented, and as long as numerical linear system solvers continue to provide high accuracy solutions) for well-posed problem instances. This second ability is typically demonstrated by asymptotic superlinear convergence properties. In this thesis, we study superlinear convergence properties of interior-point methods with proven polynomial iteration complexity. Our focus is on linear programming and semidefinite programming special cases. We provide a survey on polynomial iteration complexity interior-point methods which also achieve asymptotic superlinear convergence. We analyze the elements of superlinear convergence proofs for a dual interior-point algorithm of Nesterov and Tun\c{c}el and a primal-dual interior-point algorithm of Mizuno, Todd and Ye. We present the results of our computational experiments which observe and track superlinear convergence for a variant of Nesterov and Tun\c{c}el's algorithm

Strong Variational Sufficiency for Nonlinear Semidefinite Programming and its Implications

Strong variational sufficiency is a newly proposed property, which turns out to be of great use in the convergence analysis of multiplier methods. However, what this property implies for non-polyhedral problems remains a puzzle. In this paper, we prove the equivalence between the strong variational sufficiency and the strong second order sufficient condition (SOSC) for nonlinear semidefinite programming (NLSDP), without requiring the uniqueness of multiplier or any other constraint qualifications. Based on this characterization, the local convergence property of the augmented Lagrangian method (ALM) for NLSDP can be established under strong SOSC in the absence of constraint qualifications. Moreover, under the strong SOSC, we can apply the semi-smooth Newton method to solve the ALM subproblems of NLSDP as the positive definiteness of the generalized Hessian of augmented Lagrangian function is satisfied.Comment: 23 page

Conic Optimization: Optimal Partition, Parametric, and Stability Analysis

A linear conic optimization problem consists of the minimization of a linear objective function over the intersection of an affine space and a closed convex cone. In recent years, linear conic optimization has received significant attention, partly due to the fact that we can take advantage of linear conic optimization to reformulate and approximate intractable optimization problems. Steady advances in computational optimization have enabled us to approximately solve a wide variety of linear conic optimization problems in polynomial time. Nevertheless, preprocessing methods, rounding procedures and sensitivity analysis tools are still the missing parts of conic optimization solvers. Given the output of a conic optimization solver, we need methodologies to generate approximate complementary solutions or to speed up the convergence to an exact optimal solution. A preprocessing method reduces the size of a problem by finding the minimal face of the cone which contains the set of feasible solutions. However, such a preprocessing method assumes the knowledge of an exact solution. More importantly, we need robust sensitivity and post-optimal analysis tools for an optimal solution of a linear conic optimization problem. Motivated by the vital importance of linear conic optimization, we take active steps to fill this gap.This thesis is concerned with several aspects of a linear conic optimization problem, from algorithm through solution identification, to parametric analysis, which have not been fully addressed in the literature. We specifically focus on three special classes of linear conic optimization problems, namely semidefinite and second-order conic optimization, and their common generalization, symmetric conic optimization. We propose a polynomial time algorithm for symmetric conic optimization problems. We show how to approximate/identify the optimal partition of semidefinite optimization and second-order conic optimization, a concept which has its origin in linear optimization. Further, we use the optimal partition information to either generate an approximate optimal solution or to speed up the convergence of a solution identification process to the unique optimal solution of the problem. Finally, we study the parametric analysis of semidefinite and second-order conic optimization problems. We investigate the behavior of the optimal partition and the optimal set mapping under perturbation of the objective function vector

Numerical Solution of Optimal Control Problems with Explicit and Implicit Switches

This dissertation deals with the efficient numerical solution of switched optimal control problems whose dynamics may coincidentally be affected by both explicit and implicit switches. A framework is being developed for this purpose, in which both problem classes are uniformly converted into a mixed–integer optimal control problem with combinatorial constraints. Recent research results relate this problem class to a continuous optimal control problem with vanishing constraints, which in turn represents a considerable subclass of an optimal control problem with equilibrium constraints. In this thesis, this connection forms the foundation for a numerical treatment. We employ numerical algorithms that are based on a direct collocation approach and require, in particular, a highly accurate determination of the switching structure of the original problem. Due to the fact that the switching structure is a priori unknown in general, our approach aims to identify it successively. During this process, a sequence of nonlinear programs, which are derived by applying discretization schemes to optimal control problems, is solved approximatively. After each iteration, the discretization grid is updated according to the currently estimated switching structure. Besides a precise determination of the switching structure, it is of central importance to estimate the global error that occurs when optimal control problems are solved numerically. Again, we focus on certain direct collocation discretization schemes and analyze error contributions of individual discretization intervals. For this purpose, we exploit a relationship between discrete adjoints and the Lagrange multipliers associated with those nonlinear programs that arise from the collocation transcription process. This relationship can be derived with the help of a functional analytic framework and by interrelating collocation methods and Petrov–Galerkin finite element methods. In analogy to the dual-weighted residual methodology for Galerkin methods, which is well–known in the partial differential equation community, we then derive goal–oriented global error estimators. Based on those error estimators, we present mesh refinement strategies that allow for an equilibration and an efficient reduction of the global error. In doing so we note that the grid adaption processes with respect to both switching structure detection and global error reduction get along with each other. This allows us to distill an iterative solution framework. Usually, individual state and control components have the same polynomial degree if they originate from a collocation discretization scheme. Due to the special role which some control components have in the proposed solution framework it is desirable to allow varying polynomial degrees. This results in implementation problems, which can be solved by means of clever structure exploitation techniques and a suitable permutation of variables and equations. The resulting algorithm was developed in parallel to this work and implemented in a software package. The presented methods are implemented and evaluated on the basis of several benchmark problems. Furthermore, their applicability and efficiency is demonstrated. With regard to a future embedding of the described methods in an online optimal control context and the associated real-time requirements, an extension of the well–known multi–level iteration schemes is proposed. This approach is based on the trapezoidal rule and, compared to a full evaluation of the involved Jacobians, it significantly reduces the computational costs in case of sparse data matrices