Parametric Nonconvex Optimization via Convex Surrogates

Renzi Wang; Panagiotis Patrinos; Alberto Bemporad

math.OCApr 7, 2026

Parametric Nonconvex Optimization via Convex Surrogates

Authors: Renzi Wang, Panagiotis Patrinos, Alberto Bemporad

Organizations: IMT School for Advanced Studies, Lucca, Italy. · Department of Electrical Engineering ESAT-STADIUS, KU Leuven, Leuven, Belgium.

Abstract

This paper presents a novel learning-based approach to construct a surrogate problem that approximates a given parametric nonconvex optimization problem. The surrogate function is designed to be the minimum of a finite set of functions, given by the composition of convex and monotonic terms, so that the surrogate problem can be solved directly through parallel convex optimization. As a proof of concept, numerical experiments on a nonconvex path tracking problem confirm the approximation quality of the proposed method.

Figures & tables

Panel of Figure 1 — Fig. 1 : Level sets of the ground truth function and the surrogate functions with different numbers of components $K$ , where the dark color indicates the low function value and the yellow stars indicate the global minima.

Fig. 2 : Solutions obtained by minimizing the surrogate directly and by running L-BFGS from 100 random initializations on the original function

Fig. 3 : Illustration of the Frenet frame.

Fig. 4 : Closed-loop trajectory with different initialization methods. The trajectories of the methods cold-start , shifted solution-full , and learning-based-full are almost indistinguishable, and are denoted as converged solution for clear visualization.

Method	mean	min	max	std.
cold-start	4.140	1.857	15.565	1.063
shifted solution-full	2.582	1.467	8.686	0.567
shifted solution-2	1.717	1.151	4.426	0.382
learning-based-full (total)	2.739	1.654	5.472	0.329
learning-based-2 (total)	2.211	1.691	3.089	0.171
learning-based-full (init.)	0.618	0.504	1.230	0.087

TABLE I : Computation time (

\text{\,}\mathrm{ms}

) of different methods.

Panel of Figure 6 — Fig. 5 : Histogram of the stationarity residual of the original problem at the initial guesses and after 2 iterations of SQP, for the learning-based method and the shifted solution.

Nonconvex Optimization

Nonconvex optimization develops methods for problems whose objectives or constraints can have multiple local optima, saddle points, or nonsmooth structure.

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Abstract

Nonconvex Optimization

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