SIAM J. OPTIM. c  2013 Society for Industrial and Applied Mathematics Vol. 23, No. 2, pp. 1167–1188 A GLOBALLY CONVERGENT PROBABILITY-ONE HOMOTOPY FOR LINEAR PROGRAMS WITH LINEAR COMPLEMENTARITY CONSTRAINTS ∗ LAYNE T. WATSON † , STEPHEN C. BILLUPS ‡ , JOHN E. MITCHELL § , AND DAVID R. EASTERLING ¶ Abstract. A solution of the standard formulation of a linear program with linear complemen- tarity constraints (LPCC) does not satisfy a constraint qualification. A family of relaxations of an LPCC, associated with a probability-one homotopy map, proposed here is shown to have several desirable properties. The homotopy map is nonlinear, replacing all the constraints with nonlinear relaxations of NCP functions. Under mild existence and rank assumptions, (1) the LPCC relax- ations RLPCC(λ) have a solution for 0 ≤ λ ≤ 1; (2) RLPCC(1) is equivalent to LPCC; (3) the Kuhn–Tucker constraint qualification is satisfied at every local or global solution of RLPCC(λ) for almost all 0 ≤ λ< 1; (4) a point is a local solution of RLPCC(1) (and LPCC) if and only if it is a Kuhn–Tucker point for RLPCC(1); and (5) a homotopy algorithm can find a Kuhn–Tucker point for RLPCC(1). Since the homotopy map is a globally convergent probability-one homotopy, robust and efficient numerical algorithms exist to find solutions of RLPCC(1). Numerical results are included for some small problems. Key words. complementarity, constraint qualification, globally convergent, homotopy algo- rithm, linear program, probability-one homotopy AMS subject classifications. 65F10, 65F50, 65H10, 65K10, 90C33 DOI. 10.1137/11082868X 1. Introduction. Problems in diverse areas can be formulated as mathematical programs with complementarity constraints (MPCCs). The recent paper by Pang [39] describes applications in deregulated electricity markets, mechanical systems with fric- tional contacts, genetic regulatory networks in cell biology, control theory, and bilevel optimization. The complementarity constraints result in disjunctive mathematical programs, with feasible regions that may consist of many disjoint pieces. Linear programs with complementarity constraints (LPCCs) play an analogous role in dis- junctive programming to that of linear programs in nonlinear programming. The LPCC has many applications of its own, as surveyed by Hu, Mitchell, and Pang [23]. Complementarity constraints arise naturally in bilevel optimization, through the use of the Kuhn–Tucker conditions to express the requirement that a feasible point must solve the lower level problem. If the upper level problem is linear and if the lower level problem is either a linear program or a convex quadratic program, then ∗ Received by the editors March 28, 2011; accepted for publication (in revised form) March 7, 2013; published electronically June 10, 2013. http://www.siam.org/journals/siopt/23-2/82868.html † Departments of Computer Science and Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0106 (lwatson@computer.org). This author’s work was partially supported by National Science Foundation grant CCF-0726763, Air Force Research Laboratory grant FA8650-09-2-3938, and Air Force Office of Scientific Research grant FA9550-09-1-0153. ‡ Department of Mathematics, University of Colorado at Denver, Denver, CO 80217-3364 (stephen.Billups@ucdenver.edu). § Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180-1590 (mitchj@rpi.edu). This author’s research was partially supported by Air Force Office of Scientific Research grants FA9550-08-1-0081 and FA9550-11-1-0260. ¶ Department of Computer Science, Virginia Polytechnic Institute and State University, Blacks- burg, VA 24061-0106 (dreast@mohawk.cs.vt.edu). 1167