The Ill-Posed Linear Complementarity Problem

A regularization of the linear complementarity problem (LCP) is proposed that leads to an exact solution, if one exists, otherwise a minimizer of a natural residual of the problem is obtained. The regularized LCP (RLCP) turns out to be linear program with equilibrium constrains (LPEC) that is always solvable. For the case when the underlying matrix M of the LCP is in the class Q0 (LCP solvable if feasible), the RLCP can be solved by quadratic program, which is convex if M is positive semi-definite. An explicitly exact penalty of the RLCP formulation is also given when M E Q0 and implicitly exact otherwise. Error bounds on the distance between an arbitrary point to the set of LCP residual minimizers follow from LCP error bound theory. Computational algorithms for solving the RLCP consist of solving a convex quadratic program when M E Q0, for which a potentially finitely terminating Frank-Wolfe method is proposed. For a completely general M, a parametric method is proposed wherein for each value of the parameter a Frank-Wolfe algorithm is carried out

Second and Higher Order Duality in Nonlinear Programming

A dual problem associated with a primal nonlinear programming problem is presented that involves second derivatives of the functions constituting the primal problem. Duality results are derived for this pair of problems. More general dual problems are also presented, and duality results for these problems are also given

Error Bounds for Inconsistent Linear Inequalities and Programs

For any system of linear inequalities, consistent or not, the norm of the violations of the inequalities by a given point, multiplied by a condition constant that is independent of the point, bounds the distance between the point and the nonempty set of points that minimize these violations. Similarly, for a dual pair of possibly infeasible linear programs, the norm of violations of primal-dual feasibility and primal-dual objective equality, when multiplied by a condition constant, bounds the distance between a given point and the nonempty set of minimizers of these violations. These results extend error bounds for consistent linear inequalities and linear programs to inconsistent systems. Keywords error bounds; linear inequalities; linear programs Error bounds are playing an increasingly important role in mathematical programming. Beginning with Hoffman's classical error bound for linear inequalities [3], many papers have examined error bounds for linear and convex inequalities, line..

Nonlinear Programming Theory and Computation

A survey of nonlinear programming theory and computational algorithms is given. Subjects covered are: optimality conditions, duality theory, unconstrained and constrained optimization algorithms