Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Inexact DC Algorithms in Hilbert Spaces with Applications to PDE-Constrained Optimization

P. D. Khanh, V. V. H. Khoa, B. S. Mordukhovich, D. B. Tran, N. V. Vo

TL;DR

This work develops I-ADCA, an inexact adaptive DC algorithm for optimization in Hilbert spaces, allowing inexact subgradients and inexact subproblem solves while guaranteeing global convergence to stationary points. Under a Polyak–Łojasiewicz type property, the authors derive explicit convergence rates and extend the framework to PDE-constrained optimization with a nonconvex $L^{1-2}$ sparsity-enhanced regularizer. They formulate the PDE problem as a DC program with a strongly convex $g$ and apply a PDE-specific variant, I-ADCA-PDECO, proving well-posedness and finite-element error bounds for discretized controls. Finite element discretization yields a robust, scalable procedure with provable discretization error control, and numerical experiments confirm sparsity promotion and mesh-independent iteration behavior. Collectively, the paper advances inexact, accelerated DC methods in infinite-dimensional spaces and demonstrates their effectiveness for sparsity-promoting PDE control problems.

Abstract

In this paper, we design and apply novel inexact adaptive algorithms to deal with minimizing difference-of-convex (DC) functions in Hilbert spaces. We first introduce I-ADCA, an inexact adaptive counterpart of the well-recognized DCA (difference-of-convex algorithm), that allows inexact subgradient evaluations and inexact solutions to convex subproblems while still guarantees global convergence to stationary points. Under a Polyak-Lojasiewicz type property for DC objectives, we obtain explicit convergence rates for the proposed algorithm. Our main application addresses elliptic optimal control problems with control constraints and nonconvex $L^{1-2}$ sparsity-enhanced regularizers admitting a DC decomposition. Employing I-ADCA and appropriate versions of finite element discretization leads us to an efficient procedure for solving such problems with establishing its well-posedness and error bound estimates confirmed by numerical experiments.

Inexact DC Algorithms in Hilbert Spaces with Applications to PDE-Constrained Optimization

TL;DR

This work develops I-ADCA, an inexact adaptive DC algorithm for optimization in Hilbert spaces, allowing inexact subgradients and inexact subproblem solves while guaranteeing global convergence to stationary points. Under a Polyak–Łojasiewicz type property, the authors derive explicit convergence rates and extend the framework to PDE-constrained optimization with a nonconvex sparsity-enhanced regularizer. They formulate the PDE problem as a DC program with a strongly convex and apply a PDE-specific variant, I-ADCA-PDECO, proving well-posedness and finite-element error bounds for discretized controls. Finite element discretization yields a robust, scalable procedure with provable discretization error control, and numerical experiments confirm sparsity promotion and mesh-independent iteration behavior. Collectively, the paper advances inexact, accelerated DC methods in infinite-dimensional spaces and demonstrates their effectiveness for sparsity-promoting PDE control problems.

Abstract

In this paper, we design and apply novel inexact adaptive algorithms to deal with minimizing difference-of-convex (DC) functions in Hilbert spaces. We first introduce I-ADCA, an inexact adaptive counterpart of the well-recognized DCA (difference-of-convex algorithm), that allows inexact subgradient evaluations and inexact solutions to convex subproblems while still guarantees global convergence to stationary points. Under a Polyak-Lojasiewicz type property for DC objectives, we obtain explicit convergence rates for the proposed algorithm. Our main application addresses elliptic optimal control problems with control constraints and nonconvex sparsity-enhanced regularizers admitting a DC decomposition. Employing I-ADCA and appropriate versions of finite element discretization leads us to an efficient procedure for solving such problems with establishing its well-posedness and error bound estimates confirmed by numerical experiments.
Paper Structure (9 sections, 8 theorems, 120 equations, 4 figures, 2 tables, 3 algorithms)

This paper contains 9 sections, 8 theorems, 120 equations, 4 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Inexact Adaptive DC Algorithm in Hilbert Spaces
  4. Global Convergence and Convergence Rates of Inexact Adaptive DCA
  5. Optimal Control of Elliptic Equations with L1-2 Sparsity-Enhanced Regularization via Inexact Adaptive DCA
  6. Finite Element Discretization and Numerical Implementation
  7. Finite Element Scheme
  8. Numerical Experiments
  9. Concluding Remarks and Future Research

Key Result

Lemma 3.3

Let the function $h$ in eq:DC be continuous. At each iteration in Algorithm algo:i-adca, if $w^k$ is not a critical point critical of eq:DC, then the adaptive process terminates after a finite number of steps. Consequently, we have the estimate

Figures (4)

  • Figure 1: Desired state and optimal state output of I-ADCA-PDECO for Example \ref{['ex:pde1']}
  • Figure 2: Optimal control output and convergence rate of I-ADCA-PDECO for Example \ref{['ex:pde1']}
  • Figure 3: Optimal control output for Example \ref{['ex:pde1']} for various values of $\beta$
  • Figure 4: Desired state, optimal state, and optimal control output of I-ADCA-PDECO for Example \ref{['ex:pde2']}

Theorems & Definitions (17)

  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Theorem 3.5
  • Definition 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Theorem 4.4
  • Remark 4.5
  • ...and 7 more