A simplified nonsmooth nonconvex bundle method with applications to security-constrained ACOPF problems

TL;DR

This paper introduces a simplified bundle method for nonsmooth nonconvex optimization problems, particularly applied to security-constrained ACOPF, addressing computational challenges and providing convergence guarantees.

Contribution

It proposes a more efficient bundle algorithm inspired by SQP for nonsmooth nonconvex problems with upper-C^2 objectives, with proven convergence and practical effectiveness.

Findings

The algorithm converges globally under new assumptions.

It effectively solves security-constrained ACOPF problems.

Numerical results demonstrate improved efficiency over traditional methods.

Abstract

An optimization algorithm for a group of nonsmooth nonconvex problems inspired by two-stage stochastic programming problems is proposed. The main challenges for these problems include (1) the problems lack the popular lower-type properties such as prox-regularity assumed in many nonsmooth nonconvex optimization algorithms, (2) the objective can not be analytically expressed and (3) the evaluation of function values and subgradients are computationally expensive. To address these challenges, this report first examines the properties that exist in many two-stage problems, specifically upper-C^2 objectives. Then, we show that quadratic penalty method for security-constrained alternating current optimal power flow (SCACOPF) contingency problems can make the contingency solution functions upper-C^2 . Based on these observations, a simplified bundle algorithm that bears similarity to sequential quadratic programming (SQP) method is proposed. It is more efficient in implementation and computation compared to conventional bundle methods. Global convergence analysis of the algorithm is presented under novel and reasonable assumptions. The proposed algorithm therefore fills the gap of theoretical convergence for some smoothed SCACOPF problems. The inconsistency that might arise in our treatment of the constraints are addressed through a penalty algorithm whose convergence analysis is also provided. Finally, theoretical capabilities and numerical performance of the algorithm are demonstrated through numerical examples.