Solving Inverse optimization problems in linear programming: A Geometric and Algorithmic Approach

This paper addresses the inverse optimization problem for linear programming, focusing on determining a cost vector that ensures a pre-specified solution is optimal. Two approaches are presented: (i) using the Karush-Kuhn-Tucker (KKT) conditions, and (ii) a geometric perspective leveraging first-order necessary conditions. The latter method results in a convex quadratic programming problem, solved efficiently using the gradient projection method. Numerical experiments, including a complex resource allocation problem, validate the proposed approach. This study extends the theory and application of inverse optimization across logistics, resource management, and supply chain optimization.