<p>This paper discusses a class of two-block smooth large-scale optimization problems with both linear equality and linear inequality constraints, which have a wide range of applications, such as economic power dispatch, data mining, signal processing, etc. Our goal is to develop a novel partially feasible distributed (PFD) sequential quadratic optimization (SQO) method (PFD-SQOM) for this kind of problems. The design of the method is based on the ideas of SQO method and augmented Lagrangian Jacobi splitting scheme as well as feasible direction method, which decomposes the quadratic optimization (QO) subproblem into two small-scale QOs that can be solved independently and parallelly. A novel disturbance contraction term that can be suitably adjusted is introduced into the inequality constraints so that the feasible step size along the search direction can be increased to 1. The new iteration points are generated by the Armijo line search and the partially augmented Lagrangian function that only contains equality constraints as the merit function. The iteration points always satisfy all the inequality constraints of the problem. The global convergence and iteration complexity <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2024_2743_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$O(1/\varepsilon ^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <msup> <mi>ε</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of the proposed PFD-SQOM are obtained under appropriate assumptions without the Kurdyka–Łojasiewicz (KL) property. Furthermore, the rate of convergence such as superlinear and quadratic rates of convergence of the proposed method are analyzed when the equality constraint vanishes. Finally, the numerical effectiveness of the method is tested on a class of academic examples and an economic power dispatch problem, which shows that the proposed method is promising.</p>

A Partially Feasible Jacobi-Type Distributed SQO Method for Two-Block General Linearly Constrained Smooth Optimization

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发表日期：

2025年1月3日

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Jinbao Jian,
Wenrui Chen,
Chunming Tang,
Jianghua Yin

摘要

This paper discusses a class of two-block smooth large-scale optimization problems with both linear equality and linear inequality constraints, which have a wide range of applications, such as economic power dispatch, data mining, signal processing, etc. Our goal is to develop a novel partially feasible distributed (PFD) sequential quadratic optimization (SQO) method (PFD-SQOM) for this kind of problems. The design of the method is based on the ideas of SQO method and augmented Lagrangian Jacobi splitting scheme as well as feasible direction method, which decomposes the quadratic optimization (QO) subproblem into two small-scale QOs that can be solved independently and parallelly. A novel disturbance contraction term that can be suitably adjusted is introduced into the inequality constraints so that the feasible step size along the search direction can be increased to 1. The new iteration points are generated by the Armijo line search and the partially augmented Lagrangian function that only contains equality constraints as the merit function. The iteration points always satisfy all the inequality constraints of the problem. The global convergence and iteration complexity $O(1/\varepsilon ^2)$ $O (1 / ε^{2})$ of the proposed PFD-SQOM are obtained under appropriate assumptions without the Kurdyka–Łojasiewicz (KL) property. Furthermore, the rate of convergence such as superlinear and quadratic rates of convergence of the proposed method are analyzed when the equality constraint vanishes. Finally, the numerical effectiveness of the method is tested on a class of academic examples and an economic power dispatch problem, which shows that the proposed method is promising.