A BFGS-IP algorithm for solving strongly convex optimization problems with feasibility enforced by an exact penalty approach

• Published in: Mathematical programming Vol. 92; no. 3; pp. 393 - 424
• Main Authors: Gilbert, J. Charles, Armand, Paul, Jan-J gou, Sophie
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Heidelberg Springer 01.05.2002; Springer Nature B.V; Springer Verlag
• Subjects: Algorithms; Applied sciences; Convex analysis; Exact sciences and technology; Feasibility; Flows in networks. Combinatorial problems; Mathematical programming; Mathematics; Operational research and scientific management; Operational research. Management science; Optimization; Optimization and Control; Primal dual method; Interior point method; Problem solving; Minimization; Feasibility; Convex function; Newton method; Convex programming; Algorithm analysis; Constrained optimization; Mathematical programming; shift and slack variables; convex programming; superlinear convergence; interior point algorithm; primal-dual method; line-search; constrained optimization; BFGS quasi-Newton approximations; analytic center; infeasible iterates
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s101070100283

This paper introduces and analyses a new algorithm for minimizing a convex function subject to a finite number of convex inequality constraints. It is assumed that the Lagrangian of the problem is strongly convex. The algorithm combines interior point methods for dealing with the inequality constraints and quasi-Newton techniques for accelerating the convergence. Feasibility of the iterates is progressively enforced thanks to shift variables and an exact penalty approach. Global and q-superlinear convergence is obtained for a fixed penalty parameter; global convergence to the analytic center of the optimal set is ensured when the barrier parameter tends to zero, provided strict complementarity holds.