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<p>We are concerned with solving a generalization of the <i>p</i>-regularized subproblems (denoted by (G<i>p</i>-RS) for short). On one hand, the (G<i>p</i>-RS) with nonsingular regularized term is the main ingredient of the regularization methods for unconstrained problems and has been studied intensively. On the other hand, (G<i>p</i>-RS) itself arises in many practical applications and in those cases its regularized term may be singular so that it may have no optimal solutions. It is interesting that if the matrices in the quadratic term and in the regularized term are not simultaneously diagonalizable via congruence (SDC), the (G<i>p</i>-RS) can only be unbounded from below. If those two matrices are SDC, the (G<i>p</i>-RS) can be reformulated into one of three cases: an unbounded problem, a <i>p</i>-regularized subproblem (<i>p</i>-RS) of smaller size, or a sum of a (<i>p</i>-RS) of smaller size and an unconstrained quadratic problem with separable variables.</p>

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Solving a generalization of the p-regularized subproblems by matrix simultaneous diagonalization via congruence

  • 期刊论文

  • 发表日期：

    2025年9月17日

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  • Van-Bong Nguyen

摘要

We are concerned with solving a generalization of the p-regularized subproblems (denoted by (Gp-RS) for short). On one hand, the (Gp-RS) with nonsingular regularized term is the main ingredient of the regularization methods for unconstrained problems and has been studied intensively. On the other hand, (Gp-RS) itself arises in many practical applications and in those cases its regularized term may be singular so that it may have no optimal solutions. It is interesting that if the matrices in the quadratic term and in the regularized term are not simultaneously diagonalizable via congruence (SDC), the (Gp-RS) can only be unbounded from below. If those two matrices are SDC, the (Gp-RS) can be reformulated into one of three cases: an unbounded problem, a p-regularized subproblem (p-RS) of smaller size, or a sum of a (p-RS) of smaller size and an unconstrained quadratic problem with separable variables.