Anytime-Feasible First-Order Optimization via Safe Sequential QCQP

2025-11-24

AI Summary

This work addresses the challenge of iteratively infeasible solutions in smooth nonconvex inequality-constrained optimization. We propose the first first-order algorithm that guarantees strict feasibility at every iteration. Our method introduces a novel safety-preserving sequential QCQP dynamical systems framework, designing a vector field that ensures monotonic objective descent and forward invariance of the feasible set. Discretization is achieved via convex quadratic-constrained quadratic programming (QCQP), protective Euler integration, and an active-set strategy (SS-QCQP-AS), yielding an efficient and scalable implementation. Theoretically, the algorithm maintains an $O(1/t)$ convergence rate while ensuring all iterates strictly satisfy constraints. Empirical evaluation on multi-agent nonlinear optimal control tasks demonstrates 100% feasibility throughout optimization, convergence behavior consistent with theoretical guarantees, and solution quality competitive with second-order methods such as SQP and IPOPT.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

This paper presents the Safe Sequential Quadratically Constrained Quadratic Programming (SS-QCQP) algorithm, a first-order method for smooth inequality-constrained nonconvex optimization that guarantees feasibility at every iteration. The method is derived from a continuous-time dynamical system whose vector field is obtained by solving a convex QCQP that enforces monotonic descent of the objective and forward invariance of the feasible set. The resulting continuous-time dynamics achieve an $O(1/t)$ convergence rate to first-order stationary points under standard constraint qualification conditions. We then propose a safeguarded Euler discretization with adaptive step-size selection that preserves this convergence rate while maintaining both descent and feasibility in discrete time. To enhance scalability, we develop an active-set variant (SS-QCQP-AS) that selectively enforces constraints near the boundary, substantially reducing computational cost without compromising theoretical guarantees. Numerical experiments on a multi-agent nonlinear optimal control problem demonstrate that SS-QCQP and SS-QCQP-AS maintain feasibility, exhibit the predicted convergence behavior, and deliver solution quality comparable to second-order solvers such as SQP and IPOPT.

Problem

• Solving nonconvex optimization with continuous feasibility guarantees
• Developing first-order methods with O(1/t) convergence rates
• Enhancing scalability for constrained optimization problems

Innovation

• Safe Sequential QCQP ensures feasibility every iteration
• Safeguarded Euler discretization maintains convergence rate
• Active-set variant reduces cost without losing guarantees