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Uncomputability of Global Optima for Nonconvex Functions in the Oracle Model

K Lakshmanan

TL;DR

This work shows that, in the oracle model where a non-convex function is accessed only via finite-precision queries, neither the global minimum value nor any $\epsilon$-approximate minimizer is computable. It introduces a rigorous predicate-based framework: global optima are computable if and only if a computable predicate $Q(\zeta,x,y)$ can witness the dominance relation $f(x)\le f(y)$ via a fixed parameter $\zeta$, with Lipschitz or derivative bounds as concrete instances. The authors then identify a computable global property, the basin of attraction, under which a simple algorithm can converge to the global optimum when a bound on the basin size $m$ is known; they provide a convergence analysis and show practical viability through experiments on standard benchmarks. The results define a sharp boundary between intractability and uncomputability in oracle-based global optimization and offer a pathway for designing tractable algorithms by exploiting computable global properties. The work has implications for portfolio optimization, neural network training, and chemical process optimization by explaining fundamental limits of global optimality guarantees in realistic oracle settings.

Abstract

While it is well known that finding approximate optima of non-convex functions is computationally intractable, we show that the problem is, in fact, uncomputable in the oracle model. Specifically, we prove that no algorithm with access only to a function oracle can compute the global minimum or even an $ε$-approximation of the minimizer or minimal value. We then characterize a necessary and sufficient condition under which global optima become computable, based on the existence of a computable predicate that subsumes the global optimality condition. As an illustrative example, we consider the basin of attraction around a global minimizer as such a property and propose a simple algorithm that converges to the global minimum when a bound on the basin is known. Finally, we provide numerical experiments on standard benchmark functions to demonstrate the algorithm's practical performance.

Uncomputability of Global Optima for Nonconvex Functions in the Oracle Model

TL;DR

This work shows that, in the oracle model where a non-convex function is accessed only via finite-precision queries, neither the global minimum value nor any -approximate minimizer is computable. It introduces a rigorous predicate-based framework: global optima are computable if and only if a computable predicate can witness the dominance relation via a fixed parameter , with Lipschitz or derivative bounds as concrete instances. The authors then identify a computable global property, the basin of attraction, under which a simple algorithm can converge to the global optimum when a bound on the basin size is known; they provide a convergence analysis and show practical viability through experiments on standard benchmarks. The results define a sharp boundary between intractability and uncomputability in oracle-based global optimization and offer a pathway for designing tractable algorithms by exploiting computable global properties. The work has implications for portfolio optimization, neural network training, and chemical process optimization by explaining fundamental limits of global optimality guarantees in realistic oracle settings.

Abstract

While it is well known that finding approximate optima of non-convex functions is computationally intractable, we show that the problem is, in fact, uncomputable in the oracle model. Specifically, we prove that no algorithm with access only to a function oracle can compute the global minimum or even an -approximation of the minimizer or minimal value. We then characterize a necessary and sufficient condition under which global optima become computable, based on the existence of a computable predicate that subsumes the global optimality condition. As an illustrative example, we consider the basin of attraction around a global minimizer as such a property and propose a simple algorithm that converges to the global minimum when a bound on the basin is known. Finally, we provide numerical experiments on standard benchmark functions to demonstrate the algorithm's practical performance.
Paper Structure (17 sections, 10 theorems, 27 equations, 4 figures, 3 tables, 1 algorithm)

This paper contains 17 sections, 10 theorems, 27 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction and Preliminaries
  2. Our main contributions are as follows:
  3. Turing Machines, Computable Numbers, and Finite Precision Representation
  4. Computable Numbers
  5. Finite Precision Representation
  6. The Oracle Model
  7. Problem Definition
  8. Main Theorem
  9. Characterizing Global Optima Computability
  10. An Algorithm with Known Basin of Attraction
  11. Convergence Analysis
  12. Real-World Applications
  13. Portfolio Optimization
  14. Supervised Learning
  15. Chemical Process Optimization
  16. ...and 2 more sections

Key Result

Lemma 3.1

For all $\epsilon > 0$, there exists a finite-precision point $x^*_\epsilon$ such that where $x^* \in G^f$ is a global minimizer.

Figures (4)

  • Figure 1: The function $f$ to be minimized. Gradient descent step is shown for the interval where the function value is minimum. This interval is a subset of the basin of attraction of global minima.
  • Figure 2: Convergence to Optimum for Ackley and Rastrigin Function
  • Figure 3: Convergence to Optimum for Sphere and Rosenbrock Function
  • Figure 4: Convergence to Optimum for Beale and Booth Function

Theorems & Definitions (40)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Remark 2.9
  • Lemma 3.1
  • ...and 30 more