Anytime-Feasible First-Order Optimization via Safe Sequential QCQP

TL;DR

The paper tackles nonconvex inequality-constrained optimization with the stringent requirement that intermediate iterates remain feasible. It introduces Safe Sequential QCQP (SS-QCQP), a first-order method derived from a continuous-time dynamical system in which the search direction is produced by solving a convex QCQP that enforces descent and forward invariance, and it proves an $O(1/t)$ ergodic rate toward first-order stationarity. A safeguarded discretization with adaptive step sizes preserves feasibility and descent in discrete time, while an active-set variant SS-QCQP-AS improves scalability by restricting invariance constraints to near-active ones, maintaining the same convergence guarantees. Theoretical results establish $O(1/k)$ ergodic convergence for the discrete-time methods under standard MFCQ assumptions and Lipschitz conditions. Numerical experiments on a multi-agent control task show that SS-QCQP and SS-QCQP-AS produce feasible, competitive solutions with efficiency gains over traditional second-order solvers like SQP and IPOPT, highlighting potential for real-time safety-critical deployment in robotics and control applications.

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.

Figures (4)

• Figure 1: The search directions computed by solving the QP \ref{['eq:direction_qp']} and the QCQP \ref{['eq:direction_qcqp']} where $x = x_1, x_2^\top$, the objective $f(x)=x_2$ is a linear function and the constraint $g(x)=\|x\|_2^2-1 \leq 0$ is a unit ball constraint. Here we scale the search directions differently so that they do not overlap. When $x$ is on the boundary of the feasible region, i.e., $g(x)=0$, \ref{['eq:direction_qp']} might generate a direction tangent to the boundary, while \ref{['eq:direction_qcqp']} tilts the direction towards the interior of the feasible region.
• Figure 2: Position trajectories of four agents obtained using SS-QCQP, SS-QCQP-AS, IPOPT, and SQP.
• Figure 3: (a): Pairwise distances among the four agents over the planning horizon solved by SS-QCQP. (b): Objective of SS-QCQP, SS-QCQP-AS, IPOPT, and SQP along iterations. (c): Maximum constraint of SS-QCQP, SS-QCQP-AS, IPOPT, and SQP along iterations. (d): $\min_{i=1,...,k}\|u(x^{(i)})\|_2^2$ in SS-QCQP and $\min_{i=1,...,k}\|\hat{u}(x^{(i)})\|_2^2$ in SS-QCQP-AS along iterations.
• Figure 4: Number of active constraints in \ref{['alg:SQCQP-AS']} along the iterations. The total number of constraints is 2320.

Theorems & Definitions (27)

• Definition 1
• Definition 2: MFCQ
• Definition 3
• Lemma 1: Strict Feasibility of \ref{['eq:direction_qcqp']}
• proof
• Theorem 1
• proof
• Lemma 2
• proof
• Theorem 2