Component-wise verified solutions for constrained convex programming

Abstract

This paper presents a theorem of existence for an optimum point and methods to obtain component-wise verified solutions in constrained convex programming. The proposed method is based on the continuous Newton method and Kantorovich's theorem and slightly modifies Oishi and Tanabe's theorem for linear programming. Moreover, a theorem of existence for an optimum point for convex quadratic programming problems (CQPs) was also formulated. The CQP can be rewritten as linear complementarity problems (LCPs). A method to use component-wise verified solution method for LCPs is also presented. Numerical examples show that the proposed method is effective.