[Paper Review] A stochastic approximation method for chance-constrained nonlinear programs

Rohit Kannan, James Luedtke|arXiv (Cornell University)|Dec 17, 2018

Risk and Portfolio OptimizationDecision Sciences45 references3 citations

TL;DR

This paper proposes a stochastic approximation method for solving chance-constrained nonlinear programs by reformulating them as bi-objective problems balancing objective value and constraint violation risk. Using smoothed approximations and a projected stochastic subgradient algorithm, it converges to better approximations of the efficient frontier than sample average approximation, with consistent performance gains on three benchmark problems and a bisection method for fixed risk levels.

ABSTRACT

We propose a stochastic approximation method for approximating the efficient frontier of chance-constrained nonlinear programs. Our approach is based on a bi-objective viewpoint of chance-constrained programs that seeks solutions on the efficient frontier of optimal objective value versus risk of constraints violation. In order to be able to apply a projected stochastic subgradient algorithm to solve our reformulation with the probabilistic objective, we adapt existing smoothing-based approaches for chance-constrained problems to derive a convergent sequence of smooth approximations of our reformulated problem. In contrast with exterior sampling-based approaches (such as sample average approximation) that approximate the original chance-constrained program with one having finite support, our proposal converges to local solutions of a smooth approximation of the original problem, thereby avoiding poor local solutions that may be an artefact of a fixed sample. Computational results on three test problems from the literature indicate that our proposal is consistently able to determine better approximations of the efficient frontier than existing approaches in reasonable computation times. We also present a bisection approach for solving chance-constrained programs with a prespecified risk level.

Motivation & Objective

• To address the challenge of approximating the efficient frontier in chance-constrained nonlinear programs, where trade-offs between objective performance and constraint violation risk are critical.
• To overcome limitations of exterior sampling methods like sample average approximation, which can produce poor local solutions due to fixed sampling artifacts.
• To develop a convergent, smooth approximation scheme that enables the use of stochastic subgradient methods on the reformulated bi-objective problem.
• To provide a reliable method for solving chance-constrained programs with a prespecified risk level using a bisection approach.
• To improve computational efficiency and solution quality in approximating the efficient frontier compared to existing stochastic and sampling-based methods.

Proposed method

• Reformulates chance-constrained nonlinear programs as a bi-objective optimization problem balancing optimal objective value and risk of constraint violation.
• Applies a projected stochastic subgradient algorithm to solve the smoothed reformulation, enabling convergence to local solutions of the smooth approximation.
• Adapts existing smoothing-based techniques for chance-constrained problems to generate a sequence of smooth approximations that converge to the original problem.
• Uses a bisection algorithm to solve chance-constrained programs with a prespecified risk level, ensuring feasibility within desired probabilistic bounds.
• Employs a smoothing strategy to handle the non-smoothness of probabilistic constraints, allowing gradient-based methods to be applied effectively.
• Integrates stochastic approximation with smoothing to avoid the instability and bias inherent in fixed-sample methods like sample average approximation.

Experimental results

Research questions

• RQ1 Can a stochastic approximation method based on smoothing and subgradient optimization outperform sample average approximation in approximating the efficient frontier of chance-constrained nonlinear programs?
• RQ2 How can a bi-objective formulation of chance-constrained programs be effectively solved using stochastic subgradient methods with smooth approximations?
• RQ3 To what extent does the proposed method avoid poor local solutions that arise from fixed sampling in sample average approximation?
• RQ4 Can a bisection-based approach reliably solve chance-constrained programs with a prespecified risk level while maintaining convergence and solution quality?
• RQ5 What is the computational efficiency and solution accuracy of the proposed method compared to existing approaches on standard test problems?

Key findings

• The proposed method consistently produces better approximations of the efficient frontier than existing approaches on three benchmark test problems.
• The convergence to local solutions of a smooth approximation avoids the spurious local optima common in sample average approximation due to fixed sampling.
• Computational results show the method achieves improved solution quality within reasonable computation times, demonstrating practical efficiency.
• The bisection approach enables reliable solution of chance-constrained programs with a prespecified risk level, offering a practical tool for risk-averse optimization.
• The use of smoothing-based approximations enables the application of stochastic subgradient methods to non-smooth chance-constrained problems, enhancing algorithmic stability and convergence.
• The method demonstrates robustness and scalability on test problems, suggesting applicability to larger or more complex chance-constrained programs.