Interior-point algorithms for global optimization

We describe recent developments in interior-point algorithms for global optimization. We will focus on the algorithmic research for nonconvex quadratic programming, linear complementarity problem, and integer programming. We also outline directions in which future progress might be made.

[1]  E. Beale On quadratic proramming , 1959 .

[2]  P. Wolfe THE SIMPLEX METHOD FOR QUADRATIC PROGRAMMING , 1959 .

[3]  C. E. Lemke,et al.  Bimatrix Equilibrium Points and Mathematical Programming , 1965 .

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

[5]  Frederick S. Hillier,et al.  Efficient Heuristic Procedures for Integer Linear Programming with an Interior , 1969, Oper. Res..

[6]  B. Eaves On Quadratic Programming , 1971 .

[7]  R. Fletcher A General Quadratic Programming Algorithm , 1971 .

[8]  Arthur F. Veinott,et al.  Polyhedral sets having a least element , 1972, Math. Program..

[9]  Philip E. Gill,et al.  Newton-type methods for unconstrained and linearly constrained optimization , 1974, Math. Program..

[10]  Olvi L. Mangasarian,et al.  Linear complementarity problems solvable by A single linear program , 1976, Math. Program..

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

[12]  L. Khachiyan,et al.  The polynomial solvability of convex quadratic programming , 1980 .

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

[14]  Richard E. Stone Geometric aspects of the linear complementarity problem , 1981 .

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

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

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

[18]  G. Sonnevend An "analytical centre" for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming , 1986 .

[19]  J. Ben Rosen,et al.  An Algorithm for Global Minimization of Linearly Constrained Concave Quadratic Functions , 1987, Math. Oper. Res..

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

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

[22]  James Renegar,et al.  A polynomial-time algorithm, based on Newton's method, for linear programming , 1988, Math. Program..

[23]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988, Wiley interscience series in discrete mathematics and optimization.

[24]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

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

[26]  Shinji Mizuno,et al.  A new continuation method for complementarity problems with uniformP-functions , 1989, Math. Program..

[27]  R. Cottle,et al.  Sufficient matrices and the linear complementarity problem , 1989 .

[28]  Shinji Mizuno,et al.  A polynomial-time algorithm for a class of linear complementarity problems , 1989, Math. Program..

[29]  N. Megiddo Pathways to the optimal set in linear programming , 1989 .

[30]  Renato D. C. Monteiro,et al.  Interior path following primal-dual algorithms. part II: Convex quadratic programming , 1989, Math. Program..

[31]  D. Bayer,et al.  The Non-Linear Geometry of Linear Pro-gramming I: A?ne and projective scaling trajectories , 1989 .

[32]  Renato D. C. Monteiro,et al.  Interior path following primal-dual algorithms. part I: Linear programming , 1989, Math. Program..

[33]  Michael J. Todd,et al.  A Centered Projective Algorithm for Linear Programming , 1990, Math. Oper. Res..

[34]  Mauricio G. C. Resende,et al.  Computational experience with an interior point algorithm on the satisfiability problem , 1990, IPCO.

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

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

[37]  Jorge J. Moré,et al.  On the solution of concave knapsack problems , 1990, Math. Program..

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

[39]  Y. Ye,et al.  A Class of Linear Complementarity Problems Solvable in Polynomial Time , 1991 .

[40]  Donald Goldfarb,et al.  An O(n3L) primal interior point algorithm for convex quadratic programming , 1991, Math. Program..

[41]  Yinyu Ye,et al.  An O(n3L) potential reduction algorithm for linear programming , 1991, Math. Program..

[42]  Shinji Mizuno,et al.  An $$O(\sqrt n L)$$ iteration potential reduction algorithm for linear complementarity problems , 1991, Math. Program..

[43]  Nimrod Megiddo,et al.  An interior point potential reduction algorithm for the linear complementarity problem , 1992, Math. Program..

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