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A Block-Alternating Iterative Approach for a Class of Non-Convex Optimization Problems

Anran Li, John P. Swensen, Mehdi Hosseinzadeh

TL;DR

The paper tackles constrained non-convex optimization where the joint problem is non-convex but convex in each coordinate. It introduces a block-alternating iterative optimization (BAIO) method that sequentially optimizes each variable over convex subproblems, preserving feasibility and guaranteeing non-increasing objective values. Theoretical results establish convergence to a unique limit under strong per-coordinate convexity, with discussion on non-strongly convex cases requiring multi-start hybrid sampling to approach global minima. Empirical validation includes a numerical example and a thermal-control application, showing faster convergence and smoother control relative to GA/PSO/LQR baselines, and a publicly available Python platform (Optimization-GUI) to facilitate adoption and comparison across methods.

Abstract

Constrained non-convex optimization problems frequently arise in control applications. Solving such problems is inherently challenging, as existing methods often converge to suboptimal local minima or incur prohibitive computational costs. To address this challenge, this paper proposes a novel block-alternating iterative method that decomposes the original problem into variable-specific subproblems, which are solved iteratively. Under the assumption that the problem is convex with respect to each decision variable, the proposed approach reformulates the original problem into a sequence of convex subproblems. Theoretical results are established regarding the convergence and optimality of the method. In addition, a numerical example and a real-world control engineering application are presented to demonstrate its effectiveness. Finally, this paper introduces a ready-to-use Python platform that implements the proposed method, together with existing algorithms, to facilitate comparison and adoption.

A Block-Alternating Iterative Approach for a Class of Non-Convex Optimization Problems

TL;DR

The paper tackles constrained non-convex optimization where the joint problem is non-convex but convex in each coordinate. It introduces a block-alternating iterative optimization (BAIO) method that sequentially optimizes each variable over convex subproblems, preserving feasibility and guaranteeing non-increasing objective values. Theoretical results establish convergence to a unique limit under strong per-coordinate convexity, with discussion on non-strongly convex cases requiring multi-start hybrid sampling to approach global minima. Empirical validation includes a numerical example and a thermal-control application, showing faster convergence and smoother control relative to GA/PSO/LQR baselines, and a publicly available Python platform (Optimization-GUI) to facilitate adoption and comparison across methods.

Abstract

Constrained non-convex optimization problems frequently arise in control applications. Solving such problems is inherently challenging, as existing methods often converge to suboptimal local minima or incur prohibitive computational costs. To address this challenge, this paper proposes a novel block-alternating iterative method that decomposes the original problem into variable-specific subproblems, which are solved iteratively. Under the assumption that the problem is convex with respect to each decision variable, the proposed approach reformulates the original problem into a sequence of convex subproblems. Theoretical results are established regarding the convergence and optimality of the method. In addition, a numerical example and a real-world control engineering application are presented to demonstrate its effectiveness. Finally, this paper introduces a ready-to-use Python platform that implements the proposed method, together with existing algorithms, to facilitate comparison and adoption.
Paper Structure (11 sections, 2 theorems, 18 equations, 6 figures, 1 algorithm)

This paper contains 11 sections, 2 theorems, 18 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Block-Alternating Iterative Optimization
  3. Theoretical Properties
  4. Non-Strongly Convex Cost Functions
  5. Experimental Studies
  6. Numerical Example
  7. Thermal Control System
  8. Python Platform
  9. Conclusion
  10. Uniqueness of the Tuple $(x_1^{\dag},\cdots,x_n^{\dag})$
  11. Examination of Counterexamples Generated by ChatGPT and Copilot

Key Result

Corollary 1

The iterative structure of the method implies that, if the initial point $(x_1^{(0)},\cdots,x_n^{(0)})$ is feasible, then the updated solution $(x_1^{(k)},\cdots,x_n^{(k)})$ remains feasible for all $k \geq 0$. Moreover, the cost function $f(\cdot)$ satisfies the inequality $f(x_1^{(k)}, \dots, x_n^

Figures (6)

  • Figure 1: Geometric illustration of grid sampling, LHS, and the proposed hybrid sampling method for selecting $N=4$ initial points. The cost function is $f(x_1,x_2)=-5e^{\frac{-(x_1-1.2)^2}{0.08}}+0.02(x_1-1.2)^4+0.02(x_1-1.2)^2+10(x_2-1.2)^2+10$, and the constraints are $x_1\geq0.5$ and $x_2\leq\frac{1}{x_1-0.5}$.
  • Figure 2: Convergence of the cost function over iterations with the block-alternating iterative optimization method.
  • Figure 3: Comparison of optimization performance and computation time for the proposed block-alternating iterative method, PSO, and GA.
  • Figure 4: Experimental setup for temperature control.
  • Figure 5: Experimental results comparing the temperature control performance of the one-step-ahead predictive control method, implemented via the block-alternating iterative approach, with that of the LQR method.
  • ...and 1 more figures

Theorems & Definitions (3)

  • Corollary 1
  • Theorem 1
  • proof