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A semidefinite framework for trust region subproblems with applications to large scale minimization
 Math. Programming
, 1997
"... This is an abbreviated revision of the University of Waterloo research report CORR 9432. y ..."
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Cited by 59 (8 self)
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This is an abbreviated revision of the University of Waterloo research report CORR 9432. y
Indefinite Trust Region Subproblems And Nonsymmetric Eigenvalue Perturbations
, 1995
"... This paper extends the theory of trust region subproblems in two ways: (i) it allows indefinite inner products in the quadratic constraint and (ii) it uses a two sided (upper and lower bound) quadratic constraint. Characterizations of optimality are presented, which have no gap between necessity and ..."
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Cited by 58 (18 self)
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This paper extends the theory of trust region subproblems in two ways: (i) it allows indefinite inner products in the quadratic constraint and (ii) it uses a two sided (upper and lower bound) quadratic constraint. Characterizations of optimality are presented, which have no gap between necessity and sufficiency. Conditions for the existence of solutions are given in terms of the definiteness of a matrix pencil. A simple dual program is intro...
Interior Point Methods For Optimal Control Of DiscreteTime Systems
 Journal of Optimization Theory and Applications
, 1993
"... . We show that recently developed interior point methods for quadratic programming and linear complementarity problems can be put to use in solving discretetime optimal control problems, with general pointwise constraints on states and controls. We describe interior point algorithms for a discrete ..."
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Cited by 31 (5 self)
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. We show that recently developed interior point methods for quadratic programming and linear complementarity problems can be put to use in solving discretetime optimal control problems, with general pointwise constraints on states and controls. We describe interior point algorithms for a discrete time linearquadratic regulator problem with mixed state/control constraints, and show how it can be efficiently incorporated into an inexact sequential quadratic programming algorithm for nonlinear problems. The key to the efficiency of the interiorpoint method is the narrowbanded structure of the coefficient matrix which is factorized at each iteration. Key words. interior point algorithms, optimal control, banded linear systems. 1. Introduction. The problem of optimal control of an initial value ordinary differential equation, with Bolza objectives and mixed constraints, is min x;u Z T 0 L(x(t); u(t); t) dt + OE f (x(T )); x(t) = f(x(t); u(t); t); x(0) = x init ; (1.1) g(x(t); u(...
On a Homogeneous Algorithm for the Monotone Complementarity Problem
 Mathematical Programming
, 1995
"... We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility and compleme ..."
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Cited by 24 (3 self)
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We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility and complementarity simultaneously or a certificate proving infeasibility. Moreover, if the MCP is polynomially solvable with an interior feasible starting point, then it can be polynomially solved without using or knowing such information at all. To our knowledge, this is the first interiorpoint and infeasiblestarting algorithm for solving the MCP that possesses these desired features. Preliminary computational results are presented. Key words: Monotone complementarity problem, homogeneous and selfdual, infeasiblestarting algorithm. Running head: A homogeneous algorithm for MCP. Department of Management, Odense University, Campusvej 55, DK5230 Odense M, Denmark, email: eda@busieco.ou.dk. y De...
Iteration Algorithm for Computing Bounds in Quadratic Optimization Problems
 Complexity in Numerical Optimization
, 1993
"... We consider the problem of optimizing a quadratic function subject to integer constraints. This problem is NPhard in the general case. We present a new polynomial time algorithm for computing bounds on the solutions to such optimization problems. We transform the problem into a problem for minimizi ..."
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Cited by 6 (0 self)
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We consider the problem of optimizing a quadratic function subject to integer constraints. This problem is NPhard in the general case. We present a new polynomial time algorithm for computing bounds on the solutions to such optimization problems. We transform the problem into a problem for minimizing the trace of a matrix subject to positive definiteness condition. We then propose an interiorpoint method to solve this problem. We show that the algorithm takes no more than O(nL) iterations (where L is the the number of bits required to represent the input). The algorithm does two matrix inversions in each iteration . Keywords: Bounds, complexity, quadratic optimization, interior point methods. 1 Outline The second section of the paper shall introduce the problem of computing upper bounds on a quadratic optimization problem. We shall also motivate an interior point approach to solving the problem. The third section gives an interior point method for solving the problem. The algorith...
An Interior Point Algorithm For Linearly Constrained Optimization
 Siam J. Optim
, 1992
"... . We describe an algorithm for optimization of a smooth function subject to general linear constraints. An algorithm of the gradient projection class is used, with the important feature that the "projection" at each iteration is performed using a primaldual interior point method for convex quadrati ..."
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Cited by 2 (0 self)
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. We describe an algorithm for optimization of a smooth function subject to general linear constraints. An algorithm of the gradient projection class is used, with the important feature that the "projection" at each iteration is performed using a primaldual interior point method for convex quadratic programming. Convergence properties can be maintained even if the projection is done inexactly in a welldefined way. Higherorder derivative information on the manifold defined by the apparently active constraints can be used to increase the rate of local convergence. Key words. potential reduction algorithm, gradient porojection algorithm, linearly constrained optimization AMS(MOS) subject classifications. 65K10, 90C30 1. Introduction. We address the problem min x f(x) s.t. A T x b; (1) where x 2 R n and b 2 R m , and f is assumed throughout to be twice continuously differentiable on the level set L = fx j A T x b; f(x) f(x 0 )g; where x 0 is some given initial choice...