. This paper describes a software package, called LOQO, which implements a primaldual interior-point method for general nonlinear programming. We focus in this paper mainly on the algorithm as it applies to linear and quadratic programming with only brief mention of the extensions to convex and general nonlinear programming, since a detailed paper describing these extensions were published recently elsewhere. In particular, we emphasize the importance of establishing and maintaining symmetric quasidefiniteness of the reduced KKT system. We show that the industry standard MPS format can be nicely formulated in such a way to provide quasidefiniteness. Computational results are included for a variety of linear and quadratic programming problems. 1. INTRODUCTION LOQO is a software package for solving general (smooth) nonlinear optimization problems. It implements an infeasible-primal-dual path-following method. For linear programming, such methods were first proposed independently by Lust...

We describe a potential reduction method for convex optimization problems involving matrix inequalities. The method is based on the theory developed by Nesterov and Nemirovsky and generalizes Gonzaga and Todd's method for linear programming. A worst-case analysis shows that the number of iterations grows as the square root of the problem size, but in practice it appears to grow more slowly. As in other interior-point methods the overall computational effort is therefore dominated by the least-squares system that must be solved in each iteration. A type of conjugate-gradient algorithm can be used for this purpose, which results in important savings for two reasons. First, it allows us to take advantage of the special structure the problems often have (e.g., Lyapunov or algebraic Riccati inequalities). Second, we show that the polynomial bound on the number of iterations remains valid even if the conjugate-gradient algorithm is not run until completion, which in practice can greatly reduce the computational effort per iteration.

. In this paper, we propose primal-dual potential-reduction algorithms which can start from an infeasible interior point. We first describe two such algorithms and show that both are polynomial-time bounded. One of the algorithms decreases the Tanabe-Todd-Ye primal-dual potential function by a constant at each iteration under the condition that the duality gap decreases by at most the same ratio as the infeasibility. The other reduces a new potential function, which has one more term in the Tanabe-Todd-Ye potential function, by a fixed constant at each iteration without any other conditions on the step size. Finally, we describe modifications of these methods (incorporating centering steps) which dramatically decrease their computational complexity. Our algorithms also extend to the case of monotone linear complementarity problems. Key words. Polynomial Time, Linear Programming, Primal-Dual, Infeasible-Interior-Point Algorithm, Potential Function. AMS subject classifications. 90C05, ...

In exact arithmetic, the simplex method applied to a particular linear programming problem instance either shows that it is infeasible, shows that its dual is infeasible, or generates optimal solutions to both problems. Interiorpoint methods do not provide such clear-cut information. We provide general tools (extensions of the Farkas Lemma) for concluding that a problem or its dual is likely (in a certain well-defined sense) to be infeasible, and apply them to develop stopping rules for a generic infeasible-interior-point method and for the homogeneous self-dual algorithm for linear programming. These rules allow precise conclusions to be drawn about the linear programming problem and its dual: either near-optimal solutions are produced, or we obtain "certificates" that all optimal solutions, or all feasible solutions to the primal or dual, must have large norm. Our rules thus allow more definitive interpretation of the output of such an algorithm than previous termination criteria. We...

We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In general, constrained equations provide a unified formulation for many mathematical programming problems, including complementarity problems of various kinds and the Karush-Kuhn-Tucker systems of variational inequalities and nonlinear programs. Combining ideas from the damped Newton and interior point methods, we present an iterative algorithm for solving a constrained system of equations and investigate its convergence properties. Specialization of the algorithm and its convergence analysis to complementarity problems of various kinds and the Karush-Kuhn-Tucker systems of variational inequalities are discussed in detail. We also report the computational results of the implementation of the algo...

. Let M n (IK) denote the set of all n 2 n matrices with elements in IK, where IK represents the field IR of real numbers, the field 0 C of complex numbers or the (noncommutative) field IH of quaternion numbers. We call a subset T of M n (IK) a *-subalgebra of M n (IK) over the field IR (or simply a *-subalgebra) if (i) T forms a subring of M n (IK) with the usual addition A + B and multiplication AB of matrices A; B 2 M n (IK); specifically the zero matrix O and the identity matrix I belong to T . (ii) T is an IR-module, i.e., a vector space over the field IR; ffA + fiB 2 T for every ff; fi 2 IR and A; B 2 T , (iii) A 3 2 T if A 2 T , where A 3 denotes the conjugate transpose of A 2 M n (IK). The introduction of *-subalgebras T provides us with a unified and compact way of handling LPs (linear programs) in IR n , SDPs (semidefinite programs) in M n (IR), M n ( 0 C) and M n (IH), and monotone SDLCPs (semidefinite linear complementarity problems) in those spaces. We can extend t...

. In this paper, we study some basic properties of the monotone semidefinite nonlinear complementarity problem (SDCP). We show that the trajectory continuously accumulates into the solution set of the SDCP passing through the set of the infeasible but positive definite matrices under certain conditions. Especially, for the monotone semidefinite linear complementarity problem, the trajectory converges to an analytic center of the solution set, provided that there exists a strictly complementary solution. Finally, we propose the globally convergent infeasible-interior-point algorithm for the SDCP. Key words Monotone Semidefinite Complementarity Problem, Trajectory, Interior Point Algorithm Research Report B-312 on Mathematical and Computing Sciences, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology. 1 Introduction. Let M(n) and S(n) denote the class of n2n real matrices and the class of n2n symmetric real matrices, respectively. Assume that A; B 2 M(n)....

. Corresponding to the linear program: Maximize c T x subject to Ax = a; Bx = b; x 0; we introduce two functions in the penalty parameter t ? 0 and the Lagrange relaxation parameter vector w, ~ f p (t; w) = maxfc T x \Gamma w T (Ax \Gamma a) + t n X j=1 ln x j : Bx = b; x ? 0g (for horizontal decomposition), ~ f d (t; w) = minfa T w + b T y \Gamma t n X j=1 ln z j : B T y \Gamma z = c \Gamma A T w; z ? 0g (for vertical decomposition). For each t ? 0, ~ f p (t; \Delta) and ~ f d (t; \Delta) are strictly convex C 1 functions with a common minimizer w(t), which converges to an optimal Lagrange multiplier vector w associated with the constraint Ax = a as t ! 0, and enjoy the strong self-concordance property given by Nesterov and Nemirovsky. Based on these facts, we present conceptual algorithms with the use of Newton's method for tracing the trajectory f w(t) : t ? 0g, and analyze their computational complexity. 1. Introduction. This paper presents...

The inexact primal-dual interior point method which is discussed in this paper chooses a new iterate along an approximation to the Newton direction. The method is the Kojima, Megiddo, and Mizuno globally convergent infeasible interior point algorithm. The inexact variation takes distinct step length in both the primal and dual spaces and is globally convergent. Key Words. Linear programming, inexact primal-dual interior point algorithm, inexact search direction, short step lengths, termination criteria, global convergence 1 Introduction Consider the primal linear programming problem minimize c T x subject to: Ax = b; x 0; (1a) where A is an m-by-n matrix of full rank m, b an m-vector, and c an n-vector; and its dual problem maximize b T y subject to: A T y + z = c; z 0: (1b) Technical report number 188, Department of Informatics, University of Bergen 1 The optimality conditions for the linear program pair (1a) and (1b) are the Karush-Kuhn-Tucker (KKT) conditions: F (x;...

this paper. 1 Introduction