A complementarity problem with a continuous mapping f from Rn into itself can be written as the system of equations F(x, y) = 0 and (x, y)> 0. Here F is the mapping from R ~ " into itself defined by F(x, y) = ( xl y,, x2yZ,..., x, ~ ye, y- ffx)). Under the assumption that the mapping f is a P,,-function, we study various aspects of homotopy continuation methods that trace a trajectory consisting of solutions of the family of systems of equations F(x, y) = t(a, b) and (x, y) 8 0 until the parameter t> 0 attains 0. Here (a, b) denotes a 2n-dimensional constant positive vector. We establish the existence of a trajectory which leads to a solution of the problem, and then present a numerical method for tracing the trajectory. We also discuss the global and local convergence of the method.

This paper is a continuation of our previous paper in which we studied a polynomial primaldual path-following algorithm for SOCP using an analogue of the HRVW/KSH/M direction for SDP. We develop an improved and simplified complexity analysis which can be also applied to the algorithm using the NT direction. Specifically, we show that the long-step algorithm using the NT direction has O(n log " 01 ) iteration-complexity to reduce the duality gap by a factor of ", where n is the number of the second-order cones. The complexity for the same algorithm using the HRVW/KSH/M direction is improved to O(n 3=2 log " 01 ) from O(n 3 log " 01 ) of the previous analysis. We also show that the short and semilong-step algorithms using the NT direction (and the HRVW/KSH/M direction) have O( p n log " 01 ) and O(n log " 01 ) iterationcomplexities, respectively. keywords: second-order cone, interior-point methods, polynomial complexity, primal-dual pathfollowing methods. 1 Introduction...

We consider optimization problems where variables have either linear, or convex quadratic or semidefinite constraints. First, we define and characterize primal and dual nondegeneracy and strict complementarity conditions. Next, we develop primal-dual interior point methods for such problems and show that in the absence of degeneracy these algorithms are numerically stable. Finally we describe an implementation of our method and present numerical experiments with both degenerate and nondegenerate problems.

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 ...

Second-order cone programming (SOCP) is the problem of minimizing linear objective function over cross-section of second-order cones and an affine space. Recently this problem gets more attention because of its various important applications including quadratically constrained convex quadratic programming. In this paper we deal with a primal-dual path-following algorithm for SOCP to show many of the ideas developed for primal-dual algorithms for LP and SDP carry over to this problem. We define neighborhoods of the central trajectory in terms of the "eigenvalues" of the second-order cone, and develop an analogue of HRVW/KSH/M direction, and establish O( p n log " 01 ), O(n log " 01 ) and O(n 3 log " 01 ) iteration-complexity bounds for short-step, semilong-step and long-step path-following algorithms, respectively, to reduce the duality gap by a factor of ". keywords: second-order cone, interior-point methods, polynomial complexity, primal-dual path-following methods. 1 Intro...

This paper copes with the problem of satisfying input and/or state hard constraints in set-point tracking problems. Stability is guaranteed by synthesizing a Lyapunov quadratic function for the system, and by imposing that the terminal state lies within a level set of the function. Procedures to maximize the volume of such an ellipsoidal set are provided, and interior-point methods to solve on-line optimization are considered. Key words: Predictive control, Constraints, Lyapunov function, Set-point control, Optimization problems, Interior-point methods, Quadratically constrained quadratic programming. 1 Introduction The necessity of satisfying input/state constraints is a feature that frequently arises in control applications. Constraints are dictated for instance by physical limitations of the actuators or by the necessity to keep some plant variables within safe limits. In recent years, several control techniques have been developed which are able to handle hard constraints, see e....

A convex multiquadratic program is defined as minimizing a strictly convex quadratic function subject to convex quadratic inequality constraints. The associated Lagrangian dual problem is a strictly concave maximization problem subject to non negativity constraints. In this thesis three methods for solving the dual program are developed. The methods are based on Projected Gradient Method, Sequential Quadratic Programming, and Affine Scaling respectively. Furthermore an algorithm for solving a convex quadratic program subject to a spheric constraint, which uses Hessenberg reduction of a symmetric positive semidefinite matrix, is developed. Computational results for dense and randomly generated small to medium size convex multiquadratic programs are presented. i Acknowledgments This report constitutes my master thesis. The work presented in this thesis was conducted at the Department of Industrial and Systems Engineering (ISE), at University of Florida, Gainesville, Florida during the...

An infinite-dimensional convex optimization problem with the linear-quadratic cost function and linear-quadratic constraints is considered. We generalize the interior-point techniques of Nesterov-Nemirovsky to this infinite-dimensional situation. The obtained complexity estimates are similar to finite-dimensional ones. We apply our results to the linearquadratic control problem with quadratic constraints. It is shown that for this problem the Newton step is basically reduced to the standard LQ-problem. AMS classification: 90C20, 93C05, 49N05 Key words: control problems, quadratic constraints , path-following algorithms 1 Introduction In their fundamental monograph [9], Nesterov and Nemirovsky have shown that a broad class of a finite-dimensional optimization problems can be solved using interior-point technique based on the notion of the self-concordant barrier. Many control problems (like multi-criteria LQ and LQG problems) could be treated with the help of this technique provide...