Feasibility Evaluation of Quadratic Programs for Constrained Control

TL;DR

This work tackles online feasibility evaluation of constrained QPs in control by exploiting duality: feasibility of the primal problem is characterized by the boundedness of a properly defined dual LP. A per-configuration linear program is derived to test feasibility when soft constraints may be disregarded, enabling efficient online decision-making and constraint selection. The approach offers a theoretical and computational advantage over standard LP-based feasibility checks and demonstrates applicability to online constraint selection in CBF-QP/MPC contexts, including greedy and heuristic search strategies to maximize feasible constraint subsets. Overall, the method provides a practical, faster route to ensure feasible optimization-based controllers in fast-sampling control loops.

Abstract

This paper presents a computationally-efficient method for evaluating the feasibility of Quadratic Programs (QPs) for online constrained control. Based on the duality principle, we first show that the feasibility of a QP can be determined by the solution of a properly-defined Linear Program (LP). Our analysis yields a LP that can be solved more efficiently compared to the original QP problem, and more importantly, is simpler in form and can be solved more efficiently compared to existing methods that assess feasibility via LPs. The computational efficiency of the proposed method compared to existing methods for feasibility evaluation is demonstrated in comparative case studies as well as a feasible-constraint selection problem, indicating its promise for online feasibility evaluation of optimization-based controllers.

Figures (3)

• Figure 1: Several QP solvers compared to the proposed method. The PP (first sub-figure) the DP (second sub-figure), Chinneck's method Chinneck1996 (third sub-figure) and Proposed method \ref{['eq:dual_lp*']} (fourth sub-figure). The execution times are depicted through the colormaps in $[s]$.
• Figure 2: LP approaches compared to the proposed method. MATLAB implementations of \ref{['eq:lp_boyd']} (first sub-figure) and the proposed one \ref{['eq:dual_lp*']} (second sub-figure), GLPK implementations of \ref{['eq:lp_boyd']} (third sub-figure) and the proposed one \ref{['eq:dual_lp*']} (fourth sub-figure). The execution times are depicted through the colormaps in $[s]$.
• Figure 3: An example of a constrained QP problem over the decision variable space with hard constraints (solid lines), not-disregarded soft constraints (dashed blue lines) and disregarded soft constraints (red dashed lines) is depicted in the left figure. The arrows point to the half-plane of the respective constraint. The unconstrained optimal solution (magenta star) and the constrained optimal solution (magenta circle) are also depicted. Each soft constraint is labeled through numbers from one to five. The corresponding configuration graph is depicted in the right figure, with the feasible configurations highlighted in blue, the infeasible configurations highlighted in red and the selected configuration highlighted in magenta. For readability, the configurations are depicted as binary numbers, where "0" in the $i$-th digit denotes that the $i$-th constraint is disregarded and "1" is employed otherwise.

Theorems & Definitions (14)

• Remark 1
• Remark 2
• Theorem 1
• proof
• Definition 1: Disregarded Constraint
• Definition 2
• Remark 3
• Lemma 1
• proof
• Remark 4