On the complexity of approximating a KKT point of quadratic programming

We present a potential reduction algorithm to approximate a Karush—Kuhn—Tucker (KKT) point of general quadratic programming (QP). We show that the algorithm is a fully polynomial-time approximation scheme, and its running-time dependency on accuracy ε ∈ (0, 1) is O((l/ε) log(l/ε) log(log(l/ε))), compared to the previously best-known result O((l/ε)2). Furthermore, the limit of the KKT point satisfies the second-order necessary optimality condition of being a local minimizer. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

[1]  Katta G. Murty,et al.  Linear complementarity, linear and nonlinear programming , 1988 .

[2]  Yinyu Ye,et al.  A Fully Polynomial-Time Approximation Algorithm for Computing a Stationary Point of the General Linear Complementarity Problem , 1993, Math. Oper. Res..

[3]  G. Dantzig,et al.  COMPLEMENTARY PIVOT THEORY OF MATHEMATICAL PROGRAMMING , 1968 .

[4]  Mihalis Yannakakis,et al.  How easy is local search? , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[5]  Panos M. Pardalos,et al.  Complexity of uniqueness and local search in quadratic 0-1 programming , 1992, Oper. Res. Lett..

[6]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[7]  Philip E. Gill,et al.  Practical optimization , 1981 .

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

[9]  W. Murray.  Numerical Methods for Unconstrained Optimization , 1975 .

[10]  Yinyu Ye.  Combining Binary Search and Newton's Method to Compute Real Roots for a Class of Real Functions , 1994, J. Complex..

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

[12]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[13]  Kenneth Levenberg.  A METHOD FOR THE SOLUTION OF CERTAIN NON – LINEAR PROBLEMS IN LEAST SQUARES , 1944 .

[14]  D. Sorensen.  Newton's method with a model trust region modification , 1982 .

[15]  S. Vavasis.  Polynomial Time Weak Approximation Algorithms for Quadratic Programming , 1993 .

[16]  Panos M. Pardalos,et al.  ON THE SOLUTION OF INDEFINITE QUADRATIC PROBLEMS USING AN INTERIOR-POINT ALGORITHM , 1992 .

[17]  S. Vavasis.  Nonlinear optimization: complexity issues , 1991 .

[18]  D. Marquardt.  An Algorithm for Least-Squares Estimation of Nonlinear Parameters , 1963 .

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

[20]  Panos M. Pardalos,et al.  Constrained Global Optimization: Algorithms and Applications , 1987, Lecture Notes in Computer Science.

[21]  Jorge J. Moré,et al.  The Levenberg-Marquardt algo-rithm: Implementation and theory , 1977 .

[22]  Sartaj Sahni,et al.  Computationally Related Problems , 1974, SIAM J. Comput..

[23]  Panos M. Pardalos,et al.  Introduction to Global Optimization , 2000, Introduction to Global Optimization.

[24]  Katta G. Murty,et al.  Some NP-complete problems in quadratic and nonlinear programming , 1987, Math. Program..

[25]  Franz Rendl,et al.  A semidefinite framework for trust region subproblems with applications to large scale minimization , 1997, Math. Program..

[26]  P. Pardalos,et al.  Checking local optimality in constrained quadratic programming is NP-hard , 1988 .

[27]  Mauricio G. C. Resende,et al.  A continuous approach to inductive inference , 1992, Math. Program..

[28]  Mihir Bellare,et al.  The complexity of approximating a nonlinear program , 1995, Math. Program..

[29]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .