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Proximal-gradient algorithms for fractional programming.

Radu Ioan Boţ1,2, Ernö Robert Csetnek1

  • 1Faculty of Mathematics, University of Vienna, Vienna, Austria.

Optimization
|October 29, 2020
PubMed
Summary
This summary is machine-generated.

We developed new proximal-gradient algorithms for fractional programming. These methods efficiently find optimal solutions for problems with convex or concave denominators in real Hilbert spaces.

Keywords:
Fractional programmingKurdyka-ᴌojasiewicz propertyconvergence rateconvex subdifferentialforward–backward algorithmlimiting subdifferential

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Area of Science:

  • Optimization Theory
  • Numerical Analysis
  • Convex Analysis

Background:

  • Fractional programming problems involve optimizing ratios of functions.
  • Solving these problems is challenging due to the non-convexity or non-smoothness of the objective function.
  • Proximal-gradient methods are effective for optimization problems with non-smooth components.

Purpose of the Study:

  • To propose novel proximal-gradient algorithms for solving fractional programming problems.
  • To address problems where the numerator is proper, convex, and lower semicontinuous, and the denominator is smooth (concave or convex).
  • To analyze the convergence properties of the proposed algorithms.

Main Methods:

  • Developing two iterative algorithms combining proximal steps for the numerator and gradient steps for the denominator.
  • Applying these algorithms to fractional programming problems in real Hilbert spaces.
  • Analyzing convergence to optimal solutions or critical points based on the nature of the denominator.

Main Results:

  • The proposed algorithms effectively handle fractional programming problems with specific function properties.
  • For concave denominators, the algorithms generate sequences converging to the global optimal solution set and optimal objective value.
  • For convex denominators, the algorithms converge to the set of critical points under the Kurdyka-Łojasiewicz property.

Conclusions:

  • The introduced proximal-gradient algorithms offer efficient and convergent solutions for a class of fractional programming problems.
  • The algorithms demonstrate distinct convergence behaviors depending on whether the denominator function is concave or convex.
  • This work contributes to the advancement of optimization techniques for fractional programming in Hilbert spaces.