Quadratic programs with quadratic constraint : characterization of KKT points and equivalencewith an unconstrained problem

In this paper we consider the problem of minimizing a quadratic function with a quadratic constraint. We point out some new properties of the problem. In particular, in the rst part of the paper, we show that (i) the number of values of the objective function at KKT points is bounded by 3n + 1 where n is the dimension of the problem; (ii) given a KKT point that is not a global minimizer, it is immediate to nd a \better" feasible point; (iii) strict complementarity holds at the local-nonglobal minimum point. In the second part, we show that the original constrained problem is equivalent to the unconstrained minimization of a piecewise quartic exact merit function. Using the unconstrained formulation we give, in the nonconvex case, a new second order necessary condition for global minimimum points. A possible algorithmic application of the preceding results is brie y outlined.

[1]  M. Powell.  A method for nonlinear constraints in minimization problems , 1969 .

[2]  M. Hestenes.  Multiplier and gradient methods , 1969 .

[3]  J. Bunch,et al.  Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations , 1971 .

[4]  Torkel Glad,et al.  A multiplier method with automatic limitation of penalty growth , 1979, Math. Program..

[5]  Danny C. Sorensen,et al.  On the use of directions of negative curvature in a modified newton method , 1979, Math. Program..

[6]  David M. author-Gay.  Computing Optimal Locally Constrained Steps , 1981 .

[7]  John E. Dennis,et al.  An Adaptive Nonlinear Least-Squares Algorithm , 1977, TOMS.

[8]  Jorge J. Moré,et al.  Recent Developments in Algorithms and Software for Trust Region Methods , 1982, ISMP.

[9]  Jorge J. Moré,et al.  Computing a Trust Region Step , 1983 .

[10]  L. Grippo,et al.  A continuosly differentiable exact penalty function for nonlinear programming problems with inequality constraints , 1984, The 23rd IEEE Conference on Decision and Control.

[11]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[12]  Pravin M. Vaidya,et al.  Fast algorithms for convex quadratic programming and multicommodity flows , 1986, STOC '86.

[13]  Lanczos Algorithms for Large Symmetric Eigenvalue Computations (Jane K. Cullum And Ralph A. Willoughby) , 1987 .

[14]  L. Grippo,et al.  Exact penalty functions in constrained optimization , 1989 .

[15]  J. Dennis,et al.  Convergence theory for the structured BFGS secant method with an application to nonlinear least squares , 1989 .

[16]  Yinyu Ye,et al.  An extension of Karmarkar's projective algorithm for convex quadratic programming , 1989, Math. Program..

[17]  Richard Zippel,et al.  Proving Polynomial-Time for Sphere-Constrained Quadratic Programming , 1990 .

[18]  J. M. Martínez,et al.  Local convergence theory of inexact Newton methods based on structured least change updates , 1990 .

[19]  Thomas F. Coleman,et al.  Computing a Trust Region Step for a Penalty Function , 1990, SIAM J. Sci. Comput..

[20]  Narendra Karmarkar,et al.  An Interior-Point Approach to NP-Complete Problems , 1990, Conference on Integer Programming and Combinatorial Optimization.

[21]  Ya-Xiang Yuan,et al.  On a subproblem of trust region algorithms for constrained optimization , 1990, Math. Program..

[22]  Y. Ye,et al.  Algorithms for the solution of quadratic knapsack problems , 1991 .

[23]  Mauricio G. C. Resende,et al.  An interior point algorithm to solve computationally difficult set covering problems , 1991, Math. Program..

[24]  Ya-Xiang Yuan,et al.  A trust region algorithm for equality constrained optimization , 1990, Math. Program..

[25]  Héctor J. Martínez,et al.  Local and Superlinear Convergence for Partially Known Quasi-Newton Methods , 1991, SIAM J. Optim..

[26]  N. Karmarkar,et al.  A continuous approach to compute upper bounds in quadratic maximization problems with integer constraints , 1992 .

[27]  Matthias Heinkenschloss.  On the solution of a two ball trust region subproblem , 1992, Universität Trier, Mathematik/Informatik, Forschungsbericht.

[28]  Y. Ye.  A new complexity result on minimization of a quadratic function with a sphere constraint , 1992 .

[29]  Yinyu Ye,et al.  On affine scaling algorithms for nonconvex quadratic programming , 1992, Math. Program..

[30]  J. J. Moré.  Generalizations of the trust region problem , 1993 .

[31]  José Mario Martínez,et al.  Local Minimizers of Quadratic Functions on Euclidean Balls and Spheres , 1994, SIAM J. Optim..

[32]  Wu Li.  Differentiable Piecewise Quadratic Exact Penalty Functions for Quadratic Programs with Simple Bound Constraints , 1996, SIAM J. Optim..