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On existence of solutions to non-convex minimization problems

Rohan Rele, Angelia Nedich

TL;DR

The paper develops a unified framework to analyze the existence of minimizers for general nonconvex problems using asymptotic cones of sets and functions, introducing the notions of asymptotically bounded decay and cones of retractive directions. It derives new necessary and sufficient conditions for existence, expressed via the interplay of recession cones $X_ ext{infty}$, asymptotic cones $\mathcal{K}(f)$, and retractive cones $\mathcal{R}(X),\mathcal{R}(f)$, with refinements for structured cases including convex and polynomial constraints. The main results show that, under suitable cone inclusions (and a growth condition or regularization argument), the objective attains a finite optimum on a closed set, with the solution set being nonempty and (often) compact. The framework unifies and extends prior results on coercivity and level-set methods to nonconvex settings and provides explicit criteria for problems with functional inequalities and polynomial constraints, with implications for both finite- and infinite-dimensional spaces. Overall, the work offers tractable, geometry-based conditions to guarantee existence of solutions in broad nonconvex optimization contexts.

Abstract

We provide a unified framework for a systematic analysis of the existence of solutions to general nonconvex problems, relying on asymptotic and retractive cones for functions and sets. Using this framework we develop new necessary and sufficient conditions for the existence of solutions to a general problem of minimizing a proper closed function over a closed, possibly unbounded, set. Towards the result, we introduce cones of retractive directions for a set and a function, establishing some basic properties for them. We also investigate the relationships between the cone of retractive directions of a function and the cone of level sets of the function. Using the cones of retractive directions we provide necessary and sufficient conditions for the existence of solutions that require an asymptotically bounded decay of a function, and a relation between the cones of retractive directions of the constraint set and the asymptotic cone of the objective function. Finally we refine these conditions for more structured problems.

On existence of solutions to non-convex minimization problems

TL;DR

The paper develops a unified framework to analyze the existence of minimizers for general nonconvex problems using asymptotic cones of sets and functions, introducing the notions of asymptotically bounded decay and cones of retractive directions. It derives new necessary and sufficient conditions for existence, expressed via the interplay of recession cones , asymptotic cones , and retractive cones , with refinements for structured cases including convex and polynomial constraints. The main results show that, under suitable cone inclusions (and a growth condition or regularization argument), the objective attains a finite optimum on a closed set, with the solution set being nonempty and (often) compact. The framework unifies and extends prior results on coercivity and level-set methods to nonconvex settings and provides explicit criteria for problems with functional inequalities and polynomial constraints, with implications for both finite- and infinite-dimensional spaces. Overall, the work offers tractable, geometry-based conditions to guarantee existence of solutions in broad nonconvex optimization contexts.

Abstract

We provide a unified framework for a systematic analysis of the existence of solutions to general nonconvex problems, relying on asymptotic and retractive cones for functions and sets. Using this framework we develop new necessary and sufficient conditions for the existence of solutions to a general problem of minimizing a proper closed function over a closed, possibly unbounded, set. Towards the result, we introduce cones of retractive directions for a set and a function, establishing some basic properties for them. We also investigate the relationships between the cone of retractive directions of a function and the cone of level sets of the function. Using the cones of retractive directions we provide necessary and sufficient conditions for the existence of solutions that require an asymptotically bounded decay of a function, and a relation between the cones of retractive directions of the constraint set and the asymptotic cone of the objective function. Finally we refine these conditions for more structured problems.
Paper Structure (16 sections, 21 theorems, 123 equations, 2 figures)

This paper contains 16 sections, 21 theorems, 123 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and Terminology
  3. Asymptotic Cones
  4. Asymptotic Functions
  5. Asymptotically Bounded Decay, Cones of Retractive Directions
  6. Asymptotically Bounded Decay
  7. Cone of Retractive Directions of Sets
  8. Cone of Retractive Directions of Functions
  9. Main Results
  10. Sufficient Conditions for Existence of Solutions
  11. Conditions of Theorems \ref{['thm-compact-case']}--\ref{['main-result']} are also Necessary
  12. Implications for Convex Problems
  13. Necessary and Sufficient Conditions for Solution Existence
  14. Implications for Nonconvex Problems
  15. Generic Constraint Set
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $C_i \subseteq \mathbb{R}^n$, $i \in \mathcal{I}$, where $\mathcal{I}$ an arbitrary index set. Then, The inclusion holds as an equality for closed convex sets $C_i$.

Figures (2)

  • Figure 2.1: Set $X=\{(x,y)\in\mathbb{R}^2\mid x^2\le |y| \}$ and its asymptotic cone $X_\infty=\{(0,y)\mid y\in\mathbb{R}\}$.
  • Figure 2.2: Set $X=\{(x,y)\in\mathbb{R}^2\mid x^2\ge y \}$ and its asymptotic cone $X_\infty=\mathbb{R}^2$.

Theorems & Definitions (61)

  • Definition 2.1
  • Definition 2.2: Definition 3.3 of RockWets98
  • Proposition 2.1
  • Definition 2.3
  • Theorem 2.2
  • Definition 2.4
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Definition 3.1
  • ...and 51 more