A bicriteria perspective on L-Penalty Approaches - A corrigendum to Siddiqui and Gabriel's L-Penalty Approach for Solving MPECs

TL;DR

This paper corrects previous claims about the L-penalty method for MPECs, clarifying its limitations and conditions under which it guarantees solutions satisfying complementarity, from a bicriteria optimization perspective.

Contribution

It provides a corrigendum to prior theorems, reinterprets the L-penalty approach as a bicriteria scalarization, and specifies conditions for its effectiveness.

Findings

Revised theorems correcting previous claims.

Interpreted L-penalty as a bicriteria scalarization.

Identified assumptions for guaranteed complementarity solutions.

Abstract

This paper presents a corrigendum to Theorems 2 and 3 in Siddiqui S, Gabriel S (2013), An SOS1-Based Approach for Solving MPECs with a Natural Gas Market Application, Networks and Spatial Economics 13(2):205--227. In brief, we revise the claim that their L-penalty approach yields a solution satisfying complementarity for any positive value of L, in general. This becomes evident when interpreting the L-penalty method as a weighted-sum scalarization of a bicriteria optimization problem. We also elaborate further assumptions under which the L-penalty approach yields a solution satisfying complementarity.