<p>In a Hilbert space, we propose a class of general mixed-order primal-dual dynamical systems with Tikhonov regularization for a convex optimization problem with linear equality constraints. The proposed dynamical system is characterized by three time-dependent parameters, i.e., general viscous damping, time scaling, and Tikhonov regularization coefficients, which can incorporate as special cases some existing mixed-order primal-dual dynamical systems in the literature. With some appropriate conditions on the parameters, we analyze by constructing suitable Lyapunov functions the asymptotic convergence properties of the proposed dynamical system, where a convergence rate of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10957_2025_2798_Article_IEq1.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {O}(\frac{1}{t^2\beta (t)})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mfrac> <mn>1</mn> <mrow> <msup> <mi>t</mi> <mn>2</mn> </msup> <mi>β</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mfrac> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for the objective function error and a convergence rate of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10957_2025_2798_Article_IEq2.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$o(\frac{1}{\beta (t)})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>o</mi> <mo stretchy="false">(</mo> <mfrac> <mn>1</mn> <mrow> <mi>β</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for the primal-dual gap are established. Moreover, we further prove the strong convergence of the trajectory generated by the proposed dynamical system. Finally, we carry out some numerical experiments to illustrate the obtained theoretical results of the proposed dynamical system.</p>

A General Mixed-Order Primal-Dual Dynamical System with Tikhonov Regularization

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发表日期：

2025年7月30日

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Hong-lu Li,
Rong Hu,
Xin He,
Yi-bin Xiao

摘要

In a Hilbert space, we propose a class of general mixed-order primal-dual dynamical systems with Tikhonov regularization for a convex optimization problem with linear equality constraints. The proposed dynamical system is characterized by three time-dependent parameters, i.e., general viscous damping, time scaling, and Tikhonov regularization coefficients, which can incorporate as special cases some existing mixed-order primal-dual dynamical systems in the literature. With some appropriate conditions on the parameters, we analyze by constructing suitable Lyapunov functions the asymptotic convergence properties of the proposed dynamical system, where a convergence rate of $\mathcal {O}(\frac{1}{t^2\beta (t)})$ $O (\frac{1}{t^{2} β (t)})$ for the objective function error and a convergence rate of $o(\frac{1}{\beta (t)})$ $o (\frac{1}{β (t)})$ for the primal-dual gap are established. Moreover, we further prove the strong convergence of the trajectory generated by the proposed dynamical system. Finally, we carry out some numerical experiments to illustrate the obtained theoretical results of the proposed dynamical system.