Stochastic Sequential Quadratic Programming for Optimization with Functional Constraints

TL;DR

This paper introduces a novel stochastic sequential quadratic programming algorithm for convex optimization with nonlinear functional constraints, avoiding projections and bounded gradient assumptions, and demonstrates its efficiency through theoretical analysis and real-data experiments.

Contribution

The paper proposes the SSQP algorithm and its variants, which operate entirely in the primal domain, do not require bounded gradients, and achieve optimal oracle complexity in constrained stochastic optimization.

Findings

SSQP achieves state-of-the-art oracle complexity.

SSQP-Skip reduces computational effort by skipping quadratic sub-problems.

VARAS matches complexity bounds of unconstrained convex optimization.

Abstract

Stochastic convex optimization problems with nonlinear functional constraints are ubiquitous in signal processing applications including constrained least-squares, set-membership adaptive filtering, and trajectory optimization under uncertain fields. The presence of nonlinear functional constraints renders traditional projected stochastic gradient descent and related projection-based methods inefficient, and motivates the use of first-order methods. However, existing first-order methods, including primal and primal-dual algorithms, typically rely on a bounded (sub-)gradient assumption, which may be too restrictive in high-dimensional settings. We propose a stochastic sequential quadratic programming (SSQP) algorithm that works entirely in the primal domain, avoids projecting onto the feasible region, obviates the need for bounded gradients, and achieves state-of-the-art oracle complexity under standard smoothness and convexity assumptions. A faster version, namely SSQP-Skip, is also proposed, where the quadratic sub-problems can be skipped in most iterations. Finally, we develop an accelerated variance-reduced version of SSQP (VARAS), whose oracle complexity bounds match those for solving unconstrained finite-sum convex optimization problems. The superior performance of the proposed algorithms is demonstrated via numerical experiments on real datasets.