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HOMOTOPY CONTINUATION METHODS FOR NONLINEAR COMPLEMENTARITY PROBLEMS
, 1991
"... 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,,f ..."
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Cited by 32 (3 self)
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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 2ndimensional 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.
A Convergence Analysis of the Scalinginvariant Primaldual Pathfollowing Algorithms for Secondorder Cone Programming
 Optim. Methods Softw
, 1998
"... This paper is a continuation of our previous paper in which we studied a polynomial primaldual pathfollowing 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 di ..."
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Cited by 28 (4 self)
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This paper is a continuation of our previous paper in which we studied a polynomial primaldual pathfollowing 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 longstep algorithm using the NT direction has O(n log " 01 ) iterationcomplexity to reduce the duality gap by a factor of ", where n is the number of the secondorder 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 semilongstep 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: secondorder cone, interiorpoint methods, polynomial complexity, primaldual pathfollowing methods. 1 Introduction...
A Computational Study of the Homogeneous Algorithm for LargeScale Convex Optimization
, 1997
"... Recently the authors have proposed a homogeneous and selfdual algorithm for solving the monotone complementarity problem (MCP) [5]. The algorithm is a single phase interiorpoint type method, nevertheless it yields either an approximate optimal solution or detects a possible infeasibility of th ..."
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Cited by 13 (1 self)
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Recently the authors have proposed a homogeneous and selfdual algorithm for solving the monotone complementarity problem (MCP) [5]. The algorithm is a single phase interiorpoint 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 largescale 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, DK5230 Odense M, Denmark. Email: eda@busieco.ou.dk y ...
A Polynomial PrimalDual PathFollowing Algorithm for Secondorder Cone Programming
 Research Memorandum No. 649, The Institute of Statistical Mathematics
, 1997
"... Secondorder cone programming (SOCP) is the problem of minimizing linear objective function over crosssection of secondorder cones and an affine space. Recently this problem gets more attention because of its various important applications including quadratically constrained convex quadratic progr ..."
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Cited by 13 (1 self)
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Secondorder cone programming (SOCP) is the problem of minimizing linear objective function over crosssection of secondorder 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 primaldual pathfollowing algorithm for SOCP to show many of the ideas developed for primaldual algorithms for LP and SDP carry over to this problem. We define neighborhoods of the central trajectory in terms of the "eigenvalues" of the secondorder 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 ) iterationcomplexity bounds for shortstep, semilongstep and longstep pathfollowing algorithms, respectively, to reduce the duality gap by a factor of ". keywords: secondorder cone, interiorpoint methods, polynomial complexity, primaldual pathfollowing methods. 1 Intro...
Optimization with Semidefinite, Quadratic and Linear Constraints
 RUTCOR, RUTGERS UNIVERSITY
, 1997
"... 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 primaldual interior point methods for such problems and show ..."
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Cited by 12 (2 self)
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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 primaldual 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.
Infinite Dimensional Quadratic Optimization: InteriorPoint Methods and Control Applications
 JOURNAL OF APPLIED MATHEMATICS AND OPTIMIZATION
, 1995
"... An infinitedimensional convex optimization problem with the linearquadratic cost function and linearquadratic constraints is considered. We generalize the interiorpoint techniques of NesterovNemirovsky to this infinitedimensional situation. The obtained complexity estimates are similar to fini ..."
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Cited by 5 (3 self)
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An infinitedimensional convex optimization problem with the linearquadratic cost function and linearquadratic constraints is considered. We generalize the interiorpoint techniques of NesterovNemirovsky to this infinitedimensional situation. The obtained complexity estimates are similar to finitedimensional 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 LQproblem.
A Predictive Controller with Artificial Lyapunov Function for Linear Systems with Input/State Constraints
, 1998
"... This paper copes with the problem of satisfying input and/or state hard constraints in setpoint 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 max ..."
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Cited by 3 (0 self)
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This paper copes with the problem of satisfying input and/or state hard constraints in setpoint 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 interiorpoint methods to solve online optimization are considered. Key words: Predictive control, Constraints, Lyapunov function, Setpoint control, Optimization problems, Interiorpoint 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....
Algorithms for Convex Multiquadratic Programming
, 1997
"... 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 ..."
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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...
A RANK PRESERVING FLOW ALGORITHM FOR QUADRATIC OPTIMIZATION PROBLEMS SUBJECT TO QUADRATIC EQUALITY CONSTRAINTS
"... This paper concerns quadratic programming problems subject to quadratic equality constraints such as arise in broadband antenna array signal processing and elsewhere. At first, such a problem is converted into a semidefinite programming problem with a rank constraint. Then, a rank preserving flow is ..."
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This paper concerns quadratic programming problems subject to quadratic equality constraints such as arise in broadband antenna array signal processing and elsewhere. At first, such a problem is converted into a semidefinite programming problem with a rank constraint. Then, a rank preserving flow is used to accommodate the rank constraint. The associated gradient formulas are carefully developed. The convergence of the resulted algorithm is also guaranteed. Our approach is demonstrated by a numerical experiment. 1 PROBLEM DESCRIPTION Consider the following general quadratic problem: programming min Jo(X): = tr(XTQOX + BOX) (1) subject to: Ja(X): = tr(XTQaX + BiX) = c~, i=l,2,....?n. (2) where X E W’xq, Q. is a positive definite matrix, and Qi, i=l,2,..., m are positive semidefinite matrices. A linear constraint is covered as a special case where the matrix Qi for the corresponding index i is a zero matrix. For given generic Qi, i = 1,2,,.., m, it is a difficult task to solve the problem (1) (2). One of the main reasons is that the admissible set in the generic case is disconnected. Hence, any gradient based methods for the searching of the optimal solution is bound to lead to a local optimal. Another reason is that, eventhough one can use a gradient based method to solve it, the computation of the gradient of the cost function is complicated for problems of large size. Also ‘Partly supported by the funding of the activities of the Cooperative