Abstract. This paper gives an extensive documentation of applications of finite-dimensional nonlinear complementarity problems in engineering and equilibrium modeling. For most applications, we describe the problem briefly, state the defining equations of the model, and give functional expressions for the complementarity formulations. The goal of this documentation is threefold: (i) to summarize the essential applications of the nonlinear complementarity problem known to date, (ii) to provide a basis for the continued research on the nonlinear complementarity problem, and (iii) to supply a broad collection of realistic complementarity problems for use in algorithmic experimentation and other studies.

The formulation of interior point algorithms for semidefinite programming has become an active research area, following the success of the methods for large-- scale linear programming. Many interior point methods for linear programming have now been extended to the more general semidefinite case, but the initialization problem remained unsolved. In this paper we show that the initialization strategy of embedding the problem in a self--dual skew--symmetric problem can also be extended to the semi--definite case. This way the initialization problem of semi--definite problems is solved. This method also provides solution for the initialization of quadratic programs and it is applicable to more general convex problems with conic formulation. Key words: Semidefinite programming, complementarity, skew--symmetric embedding, initialization, self--dual problems, central path. iii 1 Introduction The extension of interior point algorithms from linear programming (LP) to semidefinite programmi...

In this paper we present several "infeasible-start" path-following and potential-reduction primal-dual interior-point methods for nonlinear conic problems. These methods try to find a recession direction of the feasible set of a self-dual homogeneous primal-dual problem. The methods under consideration generate an ffl-solution for an ffl-perturbation of an initial strictly (primal and dual) feasible problem in O( p ln fflae f ) iterations, where is the parameter of a self-concordant barrier for the cone, ffl is a relative accuracy and ae f is a feasibility measure. We also discuss the behavior of path-following methods as applied to infeasible problems. We prove that strict infeasibility (primal or dual) can be detected in O( p ln ae \Delta ) iterations, where ae \Delta is a primal or dual infeasibility measure. 1 Introduction Nesterov and Nemirovskii [9] first developed and investigated extensions of several classes of interior-point algorithms for linear programming t...

The problem of minimizing a sum of Euclidean norms dates from the 17th century and may be the earliest example of duality in the mathematical programming literature. This nonsmooth optimization problem arises in many different kinds of modern scientific applications. We derive a primal-dual interior-point algorithm for the problem, by applying Newton's method directly to a system of nonlinear equations characterizing primal and dual feasibility and a perturbed complementarity condition. The main work at each step consists of solving a system of linear equations (the Schur complement equations). This Schur complement matrix is not symmetric, unlike in linear programming. We incorporate a Mehrotra-type predictor-corrector scheme and present some experimental results comparing several variations of the algorithm, including, as one option, explicit symmetrization of the Schur complement with a skew corrector term. We also present results obtained from a code implemented to so...

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Recently the authors have proposed a homogeneous and self-dual algorithm for solving the monotone complementarity problem (MCP) [5]. The algorithm is a single phase interior-point type method, nevertheless it yields either an approximate optimal solution or detects a possible infeasibility of the problem. In this paper we specialize the algorithm to the solution of general smooth convex optimization problems that also possess nonlinear inequality constraints and free variables. We discuss an implementation of the algorithm for large-scale sparse convex optimization. Moreover, we present computational results for solving quadratically constrained quadratic programming and geometric programming problems, where some of the problems contain more than 100,000 constraints and variables. The results indicate that the proposed algorithm is also practically efficient. Department of Management, Odense University, Campusvej 55, DK-5230 Odense M, Denmark. E-mail: eda@busieco.ou.dk y ...

In this paper we introduce a conic optimization formulation for inequality-constrained convex programming, and propose a self-dual embedding model for solving the resulting conic optimization problem. The primal and dual cones in this formulation are characterized by the original constraint functions and their corresponding conjugate functions respectively. Hence they are completely symmetric. This allows for a standard primal-dual path following approach for solving the embedded problem. Moreover, there are two immediate logarithmic barrier functions for the primal and dual cones. We show that these two logarithmic barrier functions are conjugate to each other. The explicit form of the conjugate functions are in fact not required to be known in the algorithm. An advantage of the new approach is that there is no need to assume an initial feasible solution to start with. To guarantee the polynomiality of the path-following procedure, we may apply the self-concordant barrier theory of Nesterov and Nemirovski. For this purpose, as one application, we prove that the barrier functions constructed this way are indeed self-concordant when the original constraint functions are convex and quadratic. Keywords: Convex Programming, Convex Cones, Self-Dual Embedding, Self-Concordant Barrier Functions. # Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong. Research supported by Hong Kong RGC Earmarked Grants CUHK4181/00E and CUHK4233/01E. 1 1

Techniques for interactive 3D manipulation of articulated objects in cluttered environments should be geometrically aware, going beyond basic inverse or forward kinematics to allow contact while preventing interpenetration. This paper describes a general purpose interactive object manipulation technique using nonlinear optimization. The method converts geometry awareness into sets of inequality constraints and handles nonlinear equality and inequality constraints efficiently without restricting object topology. Our iterative algorithm has a quadratic convergence rate and each iteration can be solved in O(n nz (L)), where n nz (L) is the number of non-zeros in L, a Cholesky factor of a sparse matrix. To promote additional speedup, symbolic factorization is separated from numerical computation. Our approach provides a framework for using optimization techniques in interactive tools for building and manipulating models in constrained, cluttered environments. 1 Introduction Many interact...

. Two interior-point algorithms using a wide neighborhood of the central path are proposed to solve nonlinear P -complementarity problems. The proof of the polynomial complexity of the first method requires that the problem satisfies a scaled Lipschitz condition. When specialized to the monotone complementarity problems, the results of the first method are similar to the ones in Ref. 1. The second method is quite different from the first in that the proof of its global convergence does not require the scaled Lipschitz assumption. At each step of this algorithm, however, one has to compute an approximate solution of a nonlinear system such that a certain accuracy requirement is satisfied. Key Words. Interior-point algorithms, nonlinear P -complementarity problems, polynomial complexity, scaled Lipschitz condition. 2 1. Introduction Consider the complementarity problem (CP), that is, finding a pair (x; u) 2 R n \Theta R n such that u = F (x); (x; u) 0 and x T u = 0; where F ...

Most existing interior-point methods for a linear complementarity problem (LCP) require the existence of a strictly feasible point to guarantee that the iterates are bounded. Based on a regularized central path, we present an infeasible interior-point algorithm for LCPs without requiring the strict feasibility condition. The iterates generated by the algorithm are bounded when the problem is a P LCP and has a solution. Moreover, when the problem is a monotone LCP and has a solution, we prove that the convergence rate is globally linear and it achieves #-feasibility and #- complementarity in at most O(n ln(1/#)) iterations with a properly chosen starting point.