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<p>This paper concerns the tilt stability for the minimization of the sum of a twice continuously differentiable matrix-valued function and the Ky-Fan <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> <EquationSource Format="TEX">$\kappa $</EquationSource> </InlineEquation>-norm. To attain this goal, we first provide a sufficient and necessary condition for a local minimizer of the composite <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi>f</mi> <mo>=</mo> <mi>φ</mi> <mo>+</mo> <mi>g</mi> </math></EquationSource> <EquationSource Format="TEX">$f=\varphi +g$</EquationSource> </InlineEquation> to be tilt-stable with the second subderivative of <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi>g</mi> </math></EquationSource> <EquationSource Format="TEX">$g$</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <mi>g</mi> </math></EquationSource> <EquationSource Format="TEX">$g$</EquationSource> </InlineEquation> is a closed proper convex function, and <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> <EquationSource Format="TEX">$\varphi $</EquationSource> </InlineEquation> is a twice continuously differentiable function with Hessian <InlineEquation ID="IEq7"> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">∇</mi> <mn>2</mn> </msup> <mi>φ</mi> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\nabla ^{2}\varphi (\cdot )$</EquationSource> </InlineEquation> being positive semidefinite on an open convex neighborhood of the local minimizer. Then, we apply the sufficient and necessary condition to the concerned Ky-Fan <InlineEquation ID="IEq8"> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> <EquationSource Format="TEX">$\kappa $</EquationSource> </InlineEquation>-norm composite problem, and employ the expression of second subderivative of the Ky-Fan <InlineEquation ID="IEq9"> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> <EquationSource Format="TEX">$\kappa $</EquationSource> </InlineEquation>-norm to derive a verifiable criterion to identify the tilt stability of a local minimum for this class of nonconvex and nonsmooth matrix optimization. As a byproduct, a practical criterion is obtained for identifying the tilt stability of solutions to the nuclear norm and spectral norm regularized minimization problems.</p>

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Tilt Stability of Ky-Fan \(\kappa \)-Norm Composite Optimization

  • 期刊论文

  • 发表日期：

    2025年9月17日

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  • Yulan Liu,
  • Shaohua Pan,
  • Wen Song

摘要

This paper concerns the tilt stability for the minimization of the sum of a twice continuously differentiable matrix-valued function and the Ky-Fan κ $\kappa $ -norm. To attain this goal, we first provide a sufficient and necessary condition for a local minimizer of the composite f = φ + g $f=\varphi +g$ to be tilt-stable with the second subderivative of g $g$ , where g $g$ is a closed proper convex function, and φ $\varphi $ is a twice continuously differentiable function with Hessian ∇ 2 φ ( ⋅ ) $\nabla ^{2}\varphi (\cdot )$ being positive semidefinite on an open convex neighborhood of the local minimizer. Then, we apply the sufficient and necessary condition to the concerned Ky-Fan κ $\kappa $ -norm composite problem, and employ the expression of second subderivative of the Ky-Fan κ $\kappa $ -norm to derive a verifiable criterion to identify the tilt stability of a local minimum for this class of nonconvex and nonsmooth matrix optimization. As a byproduct, a practical criterion is obtained for identifying the tilt stability of solutions to the nuclear norm and spectral norm regularized minimization problems.