On the Existence of Lagrange Multipliers in Distribution Network Reconfiguration Problems

TL;DR

This paper investigates the conditions under which Lagrange multipliers exist in distribution network reconfiguration problems, proving their existence at local optima under mild assumptions, which supports the use of KKT conditions.

Contribution

It establishes the almost sure existence of Lagrange multipliers in DNR problems at local optima, clarifying theoretical foundations for optimization approaches.

Findings

Lagrange multipliers exist at local optima under mild assumptions.

KKT conditions are satisfied at these solutions.

Supports the validity of non-convex reformulations in DNR.

Abstract

Distribution network reconfiguration (DNR) is an effective approach for optimizing distribution network operation. However, the DNR problem is computationally challenging due to the mixed-integer non-convex nature. One feasible approach for addressing this challenge is to reformulate binary line on/off state variables as (continuous) non-convex constraints, leading to a nonconvex program. Unfortunately, it remains unclear whether this formulation satisfies the Karush-Kuhn-Tucker (KKT) conditions at locally optimal solutions. In this brief, we study the existence of Lagrange multipliers in DNR problems and prove that under mild assumptions, Lagrange multipliers exist for the DNR model at every locally optimal solution almost surely such that the KKT conditions hold.