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Global Solutions to Non-Convex Functional Constrained Problems with Hidden Convexity

Ilyas Fatkhullin, Niao He, Guanghui Lan, Florian Wolf

TL;DR

This work addresses non-convex constrained optimization with hidden convexity, where a consistent nonlinear transformation $c$ links a non-convex problem to a convex reformulation. It introduces two global-convergence algorithms that operate in the original space using only (sub-)gradients: an Inexact Proximal Point Method (IPPM) with a two-level constraint shift to guarantee Slater conditions, and a bundle-level approach with shifted linearizations to avoid relying on Slater. The main contributions are the first non-asymptotic, provable global-minimization guarantees for hidden-convex constrained problems without constraint qualifications, including explicit oracle-complexity bounds in both non-smooth and smooth settings, and variants that handle known/unknown optimal values through shifted bundle techniques. The results unify and extend classical convex-optimization tools to the hidden-convex setting, achieving complexities matching unconstrained hidden-convex optimization and enabling scalable solutions for applications in control, reinforcement learning, and geometric programming without CQ assumptions.

Abstract

Constrained non-convex optimization is fundamentally challenging, as global solutions are generally intractable and constraint qualifications may not hold. However, in many applications, including safe policy optimization in control and reinforcement learning, such problems possess hidden convexity, meaning they can be reformulated as convex programs via a nonlinear invertible transformation. Typically such transformations are implicit or unknown, making the direct link with the convex program impossible. On the other hand, (sub-)gradients with respect to the original variables are often accessible or can be easily estimated, which motivates algorithms that operate directly in the original (non-convex) problem space using standard (sub-)gradient oracles. In this work, we develop the first algorithms to provably solve such non-convex problems to global minima. First, using a modified inexact proximal point method, we establish global last-iterate convergence guarantees with $\widetilde{\mathcal{O}}(\varepsilon^{-3})$ oracle complexity in non-smooth setting. For smooth problems, we propose a new bundle-level type method based on linearly constrained quadratic subproblems, improving the oracle complexity to $\widetilde{\mathcal{O}}(\varepsilon^{-1})$. Surprisingly, despite non-convexity, our methodology does not require any constraint qualifications, can handle hidden convex equality constraints, and achieves complexities matching those for solving unconstrained hidden convex optimization.

Global Solutions to Non-Convex Functional Constrained Problems with Hidden Convexity

TL;DR

This work addresses non-convex constrained optimization with hidden convexity, where a consistent nonlinear transformation links a non-convex problem to a convex reformulation. It introduces two global-convergence algorithms that operate in the original space using only (sub-)gradients: an Inexact Proximal Point Method (IPPM) with a two-level constraint shift to guarantee Slater conditions, and a bundle-level approach with shifted linearizations to avoid relying on Slater. The main contributions are the first non-asymptotic, provable global-minimization guarantees for hidden-convex constrained problems without constraint qualifications, including explicit oracle-complexity bounds in both non-smooth and smooth settings, and variants that handle known/unknown optimal values through shifted bundle techniques. The results unify and extend classical convex-optimization tools to the hidden-convex setting, achieving complexities matching unconstrained hidden-convex optimization and enabling scalable solutions for applications in control, reinforcement learning, and geometric programming without CQ assumptions.

Abstract

Constrained non-convex optimization is fundamentally challenging, as global solutions are generally intractable and constraint qualifications may not hold. However, in many applications, including safe policy optimization in control and reinforcement learning, such problems possess hidden convexity, meaning they can be reformulated as convex programs via a nonlinear invertible transformation. Typically such transformations are implicit or unknown, making the direct link with the convex program impossible. On the other hand, (sub-)gradients with respect to the original variables are often accessible or can be easily estimated, which motivates algorithms that operate directly in the original (non-convex) problem space using standard (sub-)gradient oracles. In this work, we develop the first algorithms to provably solve such non-convex problems to global minima. First, using a modified inexact proximal point method, we establish global last-iterate convergence guarantees with oracle complexity in non-smooth setting. For smooth problems, we propose a new bundle-level type method based on linearly constrained quadratic subproblems, improving the oracle complexity to . Surprisingly, despite non-convexity, our methodology does not require any constraint qualifications, can handle hidden convex equality constraints, and achieves complexities matching those for solving unconstrained hidden convex optimization.

Paper Structure

This paper contains 21 sections, 8 theorems, 49 equations, 3 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Constrained Optimization.
  4. Constraint Qualifications (CQs).
  5. Notations and Standard Assumptions
  6. Hidden Convex Problem Class
  7. Relaxing invertibility condition.
  8. Globally Optimal Solutions and KKT Points
  9. Non-convex Equality Constraints
  10. NP-Hardness under Inconsistent Transformation
  11. Motivating Applications
  12. Geometric Programming eckerGeometricProgrammingMethods1980boydTutorialGeometricProgramming2007xiaSurveyHiddenConvex2020.
  13. Convex Constrained Markov Decision Process (CCMPD) zhangVariationalPolicyGradient2020.
  14. Controller Synthesis in Optimal Control.
  15. Observations for Convergence Analysis
  16. ...and 6 more sections

Key Result

proposition thmcounterproposition

Let $F_i(\,\cdot\,{})$ be weakly convex on ${\mathcal{X}}$ for $i = 1, 2$ and problem eq:MainProblem be hidden convex. Assume $c(\,\cdot\,{})$ is differentiable for some $\hat{{x}} \in {\mathcal{X}}$. Then the following implications hold:

Figures (3)

  • Figure 1: Illustrative examples of non-smooth (top) and smooth (bottom) hidden convex problems: (a) and (b) -- Constrained Non-Linear Least Squares; (c) and (d) -- Constrained Geometric Programming, see \ref{['eq:ToyExampleLeastSquares']} and \ref{['eq:ToyExampleGeometricProgramming']} in \ref{['subsec:ExampleNonsmoothConstrainedLeastSquares']} for details. The plots illustrate in color the level sets of the non-convex formulation (left) and the convex formulation (right). The feasible sets are shown as gray regions: $\{F_2 \leq 0\}$ in the ${\mathcal{X}}$-domain, and $\{H_2 \leq 0\}$ in the ${\mathcal{U}}$-domain. We use the notation ${x}^{*}_\mathrm{uncon}$ to denote the optimum of $\min_{x} F_1({x})$ without constraints and ${x}^{*}$ the minimizer under constraints; analogously, ${u}^{*}_\mathrm{uncon}$ and ${u}^{*}$ denote their counterparts in ${\mathcal{U}}$.
  • Figure 2: An illustrative example of a hidden convex problem with inconsistent transformations\ref{['eq:IntroEquationInconsistent_Transform']}.The grey regions illustrate the feasible set, and the objective value is shown in color. This problem has two local minima: the sub-optimal leftmost point ${x}_{\mathrm{local}} = (-0.3, 0.2)^\top$, and the optimal rightmost point ${x}^{*} = (0.5, 1.5);$${x}^{*}_\mathrm{uncon}$ denotes the global optimum without constraints. A local search method may terminate at the sub-optimal local minima ${x}_{\mathrm{local}}.$
  • Figure 3: (a) and (c): To illustrate the need for the shift, we use the hidden convex function $F_1({x}) \coloneqq 1 - \cos(\pi \cdot {x})$ without constraints. (b): We illustrate the shifted constraint on the constrained geometric programming example $F_1({x}) \coloneqq {x}_1 \cdot {x}_2 + \frac{4}{{x}_1} + \frac{1}{{x}_2}$ constrained to $\{F_2 \leq 0\}$, where $F_2({x}) \coloneqq {x}_1 \cdot {x}_2 - 1$, cf. \ref{['eq:ToyExampleGeometricProgramming']} in \ref{['subsec:ExampleNonsmoothConstrainedLeastSquares']}.

Theorems & Definitions (20)

  • definition thmcounterdefinition: Hidden Convexity
  • definition thmcounterdefinition
  • definition thmcounterdefinition: Slater's Condition
  • proposition thmcounterproposition
  • proof
  • proposition thmcounterproposition
  • proof
  • proposition thmcounterproposition: fatkhullinStochasticOptimizationHidden2024 Prop. 3
  • lemma thmcounterlemma: HC--Slater's Lemma
  • proof
  • ...and 10 more