Cuts for mixed 0-1 conic programming

In this we paper we study techniques for generating valid convex constraints for mixed 0-1 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 0-1 linear programs, such as the Gomory cuts, the lift-and-project cuts, and cuts from other hierarchies of tighter relaxations, extend in a straightforward manner to mixed 0-1 conic programs. We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts. Gomory cuts for mixed 0-1 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems, e.g. we show that all the subtour elimination inequalities for the traveling salesman problem are rank-1 Gomory cuts with respect to a single semidefinite constraint. We also include results from our preliminary computational experiments with these cuts.

[1]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[2]  Arkadi Nemirovski,et al.  Robust Modeling of Multi-Stage Portfolio Problems , 2000 .

[3]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

[4]  Gregory Gutin,et al.  The traveling salesman problem , 2006, Discret. Optim..

[5]  Warren P. Adams,et al.  A hierarchy of relaxation between the continuous and convex hull representations , 1990 .

[6]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems , 1990, SIAM J. Discret. Math..

[7]  Martin W. P. Savelsbergh,et al.  An Updated Mixed Integer Programming Library: MIPLIB 3.0 , 1998 .

[8]  Ali Ridha Mahjoub,et al.  On the cut polytope , 1986, Math. Program..

[9]  Sanjay Mehrotra,et al.  A branch-and-cut method for 0-1 mixed convex programming , 1999, Math. Program..

[10]  John E. Mitchell,et al.  Computational Experience with an Interior Point Cutting Plane Algorithm , 1999, SIAM J. Optim..

[11]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations and Convex Hull Characterizations for Mixed-integer Zero-one Programming Problems , 1994, Discret. Appl. Math..

[12]  Howard J. Karloff.  How Good is the Goemans-Williamson MAX CUT Algorithm? , 1999, SIAM J. Comput..

[13]  Robert J. Vanderbei,et al.  Solving Problems with Semidefinite and Related Constraints Using Interior-Point Methods for Nonlinear Programming , 2003, Math. Program..

[14]  John E. Mitchell,et al.  Restarting after Branching in the SDP Approach to MAX-CUT and Similar Combinatorial Optimization Problems , 2001, J. Comb. Optim..

[15]  Donald Goldfarb,et al.  Robust Portfolio Selection Problems , 2003, Math. Oper. Res..

[16]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[17]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[18]  Michel X. Goemans,et al.  Semideenite Programming in Combinatorial Optimization , 1999 .

[19]  Vladimir A. Yakubovich,et al.  Linear Matrix Inequalities in System and Control Theory (S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan) , 1995, SIAM Rev..

[20]  C. Helmberg,et al.  Solving quadratic (0,1)-problems by semidefinite programs and cutting planes , 1998 .

[21]  Stephen P. Boyd,et al.  Portfolio optimization with linear and fixed transaction costs , 2007, Ann. Oper. Res..

[22]  J. Mitchell,et al.  Solving Linear Ordering Problems with a Combined Interior Pointtsimplex Cutting Plane Algorithm * , 1999 .

[23]  Gerhard Reinelt,et al.  TSPLIB - A Traveling Salesman Problem Library , 1991, INFORMS J. Comput..

[24]  Petersen CliffordC..  A Capital Budgeting Heuristic Algorithm Using Exchange Operations , 1974 .

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

[26]  Christoph Helmberg.  A Cutting Plane Algorithm for Large Scale Semidefinite Relaxations , 2004, The Sharpest Cut.

[27]  Egon Balas,et al.  A lift-and-project cutting plane algorithm for mixed 0–1 programs , 1993, Math. Program..

[28]  Arkadi Nemirovski,et al.  Robust Convex Optimization , 1998, Math. Oper. Res..

[29]  Mirjana Cangalovic,et al.  Semidefinite Programming Methods for the Symmetric Traveling Salesman Problem , 1999, IPCO.

[30]  Sanjay Mehrotra,et al.  Generating Convex Polynomial Inequalities for Mixed 0–1 Programs , 2002, J. Glob. Optim..

[31]  Laurent El Ghaoui,et al.  Robust Solutions to Uncertain Semidefinite Programs , 1998, SIAM J. Optim..

[32]  Farid Alizadeh,et al.  Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization , 1995, SIAM J. Optim..

[33]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[34]  William J. Cook,et al.  Combinatorial optimization , 1997 .

[35]  Monique Laurent,et al.  A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0-1 Programming , 2003, Math. Oper. Res..

[36]  Clifford C. Petersen,et al.  Computational Experience with Variants of the Balas Algorithm Applied to the Selection of R&D Projects , 1967 .

[37]  R. Bixby,et al.  On the Solution of Traveling Salesman Problems , 1998 .

[38]  Henry Wolkowicz,et al.  Strengthened semidefinite relaxations via a second lifting for the Max-Cut problem , 2002, Discret. Appl. Math..

[39]  M. R. Rao,et al.  Combinatorial Optimization , 1992, NATO ASI Series.

[40]  John E. Mitchell,et al.  Fixing variables and generating classical cutting planes when using an interior point branch and cut method to solve integer programming problems , 1997 .

[41]  Howard J. Karloff,et al.  How good is the Goemans-Williamson MAX CUT algorithm? , 1996, STOC '96.

[42]  Donald Goldfarb,et al.  Second-order cone programming , 2003, Math. Program..

[43]  Alexander Schrijver,et al.  Cones of Matrices and Set-Functions and 0-1 Optimization , 1991, SIAM J. Optim..