Pricing Discrete and Nonlinear Markets With Semidefinite Relaxations

2026-02-17

AI Summary

This work addresses the challenge of efficient marginal pricing in non-convex electricity markets characterized by discrete decisions and nonlinear constraints, which often necessitate out-of-market subsidies. The authors propose a novel approach that integrates semidefinite programming (SDP) relaxation with the envelope theorem to derive marginal price signals from the dual variables of the relaxed problem, offering theoretical guarantees on their economic coherence. The associated opportunity cost is rigorously bounded by the SDP relaxation gap. The method is applicable to non-convex market models incorporating both DC and AC unit commitment formulations. Numerical experiments on IEEE benchmark instances demonstrate that the proposed pricing mechanism reduces average opportunity costs by 46% compared to conventional fixed-binary pricing schemes, while the SDP relaxation is typically tight in practice.

Technology Category

• Search and Optimization: Non-convex Optimization
• Game Theory and Economic Paradigms: Mechanism Design
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Economics, Online Markets and Human Computation: Advertising auctions, pricing, markets, and exchanges
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments

Abstract

Nonconvexities in markets with discrete decisions and nonlinear constraints make efficient pricing challenging, often necessitating subsidies. A prime example is the unit commitment (UC) problem in electricity markets, where costly subsidies are commonly required. We propose a new pricing scheme for nonconvex markets with both discreteness and nonlinearity, by convexifying nonconvex structures through a semidefinite programming (SDP) relaxation and deriving prices from the relaxation's dual variables. When the choice set is bounded, we establish strong duality for the SDP, which allows us to extend the envelope theorem to the value function of the relaxation. This extension yields a marginal price signal for demand, which we use as our pricing mechanism. We demonstrate that under certain conditions-for instance, when the relaxation's right hand sides are linear in demand-the resulting lost opportunity cost is bounded by the relaxation's optimality gap. This result highlights the importance of achieving tight relaxations. The proposed framework applies to nonconvex electricity market problems, including for both direct current and alternating current UC. Our numerical experiments indicate that the SDP relaxations are often tight, reinforcing the effectiveness of the proposed pricing scheme. Across a suite of IEEE benchmark instances, the lost opportunity cost under our pricing scheme is, on average, 46% lower than that of the commonly used fixed-binary pricing scheme.

Problem

• nonconvex markets
• discrete decisions
• nonlinear constraints
• pricing
• unit commitment

Innovation

• semidefinite programming
• nonconvex pricing
• unit commitment
• strong duality
• envelope theorem