Characterizing Shadow Price via Lagrangian Multiplier for Nonsmooth Problem

TL;DR

This paper explores the relationship between shadow prices and Lagrangian multipliers in nonsmooth convex and Lipschitzian optimization problems, providing theoretical insights and an application to electricity pricing.

Contribution

It establishes that Lagrangian multipliers serve as upper bounds for shadow prices in nonsmooth convex and Lipschitzian optimization problems, advancing shadow pricing methods.

Findings

Lagrangian multiplier bounds shadow price in nonsmooth convex problems

The paper lists several nonsmooth functions in Lipschitzian optimization

Application to electricity pricing demonstrates practical relevance

Abstract

In this paper, a relation between shadow price and the Lagrangian multiplier for nonsmooth problem is explored. It is shown that the Lagrangian Multiplier is the upper bound of shadow price for convex optimization and a class of Lipschtzian optimizations. This work can be used in shadow pricing for nonsmooth situation. The several nonsmooth functions involved in this class of Lipschtzian optimizations is listed. Finally, an application to electricity pricing is discussed.