<p>This paper analyzes a proximal gradient method for nondifferentiable convex multiobjective optimization problems, where the components of the objective function are the sum of a proper lower semicontinuous function and a continuously differentiable function. By adopting a typical line search procedure, it is found that without a Lipschitz continuity of the gradients of the smooth part of the objective function, the proposed method is able to generate sequences that converge to weakly Pareto optimal points. The convergence rate of the method is found to be <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10589_2025_663_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {O}(1/k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Further, if the smooth component in the objective function is strongly convex, then the convergence rate is <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10589_2025_663_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {O}(r^k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>r</mi> <mi>k</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10589_2025_663_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$r\in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Moreover, in the absence of a strong convexity assumption, we also consider the accelerated version of the proposed approach based on the Nesterov step strategy. We obtain the improved convergence rate of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10589_2025_663_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {O}(1/k^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <msup> <mi>k</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, which is measured by a merit function. Numerical implementation strategies and performance profiles of the proposed methods on the considered problem involving <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10589_2025_663_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-norm and indicator function are also provided.</p>

Proximal gradient method for convex multiobjective optimization problems without Lipschitz continuous gradients

期刊论文
发表日期：

2025年2月10日

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Xiaopeng Zhao,
Ravi Raushan,
Debdas Ghosh,
Jen-Chih Yao,
Min Qi

摘要

This paper analyzes a proximal gradient method for nondifferentiable convex multiobjective optimization problems, where the components of the objective function are the sum of a proper lower semicontinuous function and a continuously differentiable function. By adopting a typical line search procedure, it is found that without a Lipschitz continuity of the gradients of the smooth part of the objective function, the proposed method is able to generate sequences that converge to weakly Pareto optimal points. The convergence rate of the method is found to be $\mathcal {O}(1/k)$ $O (1 / k)$ . Further, if the smooth component in the objective function is strongly convex, then the convergence rate is $\mathcal {O}(r^k)$ $O (r^{k})$ for some $r\in (0,1)$ $r \in (0, 1)$ . Moreover, in the absence of a strong convexity assumption, we also consider the accelerated version of the proposed approach based on the Nesterov step strategy. We obtain the improved convergence rate of $\mathcal {O}(1/k^2)$ $O (1 / k^{2})$ , which is measured by a merit function. Numerical implementation strategies and performance profiles of the proposed methods on the considered problem involving $\ell _1$ $ℓ_{1}$ -norm and indicator function are also provided.