A Surrogate Objective Framework for Prediction+Optimization with Soft Constraints

TL;DR

This paper introduces a differentiable surrogate objective framework for prediction+optimization problems with soft constraints, enabling better alignment between prediction training and optimization goals in real-world applications.

Contribution

It presents a novel framework that handles soft constraints directly, providing theoretical bounds and closed-form solutions for gradients in linear and quadratic programming.

Findings

Outperforms traditional two-stage methods in experiments

Effective in portfolio optimization and resource provisioning

Handles soft constraints with max operators successfully

Abstract

Prediction+optimization is a common real-world paradigm where we have to predict problem parameters before solving the optimization problem. However, the criteria by which the prediction model is trained are often inconsistent with the goal of the downstream optimization problem. Recently, decision-focused prediction approaches, such as SPO+ and direct optimization, have been proposed to fill this gap. However, they cannot directly handle the soft constraints with the $max$ operator required in many real-world objectives. This paper proposes a novel analytically differentiable surrogate objective framework for real-world linear and semi-definite negative quadratic programming problems with soft linear and non-negative hard constraints. This framework gives the theoretical bounds on constraints' multipliers, and derives the closed-form solution with respect to predictive parameters and thus gradients for any variable in the problem. We evaluate our method in three applications extended with soft constraints: synthetic linear programming, portfolio optimization, and resource provisioning, demonstrating that our method outperforms traditional two-staged methods and other decision-focused approaches.