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Convolutional optimization with convex kernel and power lift

Zhipeng Lu

TL;DR

The paper proposes a deterministic global optimization framework based on convolving the objective with a convex kernel and applying a power lift, e.g., $\varphi(f)=f^N$, to concentrate mass near the maximum. It establishes a theoretical foundation showing convexity preservation under convolution with nonnegative functions, introduces a family of convex kernels, and develops integral-concentration and restriction principles to guide search directions, including directional convolutions $g\ast_{(a,v)} f$. Two practical models are developed: a one-dimensional convolutional model $F_{\delta,N}=g_{\delta}*f^N$ and its higher-dimensional generalization, plus zigzag variants that reduce dimensionality by alternating one-dimensional searches. The approach is complemented by preliminary computational results on standard test functions (e.g., LOG1/LOG2, POLY1, Rosenbrock), illustrating feasibility while acknowledging limited scope. Collectively, the work aims to provide a robust, deteministic alternative for locating global optima in settings where statistical methods are inadequate, with potential benefits for high-dimensional optimization tasks where guarantees and stability are desired.

Abstract

We focus on establishing the foundational paradigm of a novel optimization theory based on convolution with convex kernels. Our goal is to devise a morally deterministic model of locating the global optima of an arbitrary function, which is distinguished from most commonly used statistical models. Limited preliminary numerical results are provided to test the efficiency of some specific algorithms derived from our paradigm, which we hope to stimulate further practical interest.

Convolutional optimization with convex kernel and power lift

TL;DR

The paper proposes a deterministic global optimization framework based on convolving the objective with a convex kernel and applying a power lift, e.g., , to concentrate mass near the maximum. It establishes a theoretical foundation showing convexity preservation under convolution with nonnegative functions, introduces a family of convex kernels, and develops integral-concentration and restriction principles to guide search directions, including directional convolutions . Two practical models are developed: a one-dimensional convolutional model and its higher-dimensional generalization, plus zigzag variants that reduce dimensionality by alternating one-dimensional searches. The approach is complemented by preliminary computational results on standard test functions (e.g., LOG1/LOG2, POLY1, Rosenbrock), illustrating feasibility while acknowledging limited scope. Collectively, the work aims to provide a robust, deteministic alternative for locating global optima in settings where statistical methods are inadequate, with potential benefits for high-dimensional optimization tasks where guarantees and stability are desired.

Abstract

We focus on establishing the foundational paradigm of a novel optimization theory based on convolution with convex kernels. Our goal is to devise a morally deterministic model of locating the global optima of an arbitrary function, which is distinguished from most commonly used statistical models. Limited preliminary numerical results are provided to test the efficiency of some specific algorithms derived from our paradigm, which we hope to stimulate further practical interest.

Paper Structure

This paper contains 20 sections, 14 theorems, 45 equations, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Theoretical basics
  3. Convolution preserving convexity
  4. Critical point approximation
  5. Construction of convex kernel
  6. Integral concentration
  7. Restriction of convex functions
  8. Restricted direction on unit sphere
  9. Non-restricted direction
  10. Models of convolutional optimization
  11. One-dimensional model
  12. Higher-dimensional model
  13. One-dimensional "zigzag" variant algorithms for higher dimensional optimization
  14. Computational results de facto
  15. Disclaimer about limited experimental results
  16. ...and 5 more sections

Key Result

Lemma 1

Suppose $f(x)\geq 0$ is continuous and $g(x)$ is convex. Then their convolution $g\ast f(x)$ is convex.

Theorems & Definitions (21)

  • Lemma 1
  • proof
  • Corollary 1
  • Lemma 2
  • proof
  • Corollary 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 11 more