A Globally Convergent Probability-One Homotopy for Linear Programs with Linear Complementarity Constraints

A solution of the standard formulation of a linear program with linear complementarity 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 relaxations 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.