arXiv subjects

Xiaoxi Jia

Publications and source records attributed to Xiaoxi Jia.

7 recordsLinked to original sources

arXiv2026

A constraint dissolving inexact penalty method for optimization problems with geometric constraints

Optimization problems with geometric constraints have a broad range of applications, including machine learning, finance, and control. A powerful algorithmic tool to resolve these geometric constraints are constraint dissolving methods. To this end, we propose a framework for constraint dissolving mappings for nonconvex geometric constraints. Leveraging these, we develop a constraint dissolving inexact penalty method to solve optimization problems with general set-membership constraints and possibly nonconvex geometric constraints. We establish the convergence of the proposed algorithm and prove that every feasible accumulation point is Mordukhovich stationary. Notably, we rely only on mild asymptotic Mordukhovich regularity, which is significantly weaker than the constraint qualifications adopted in the existing literature on constraint dissolving methods. Numerical experiments addressing classical equality-, complementarity-, sparsity-, and low-rank constrained optimization problems demonstrate that the proposed method is competitive with the safeguarded augmented Lagrangian method in terms of solution quality and significantly outperforms the penalty decomposition method.

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arXiv2025

Projection-based curve pattern search for black-box optimization over smooth convex sets

In this paper, we deal with the problem of optimizing a black-box smooth function over a full-dimensional smooth convex set. We study sets of feasible curves that allow to properly characterize stationarity of a solution and possibly carry out sound backtracking curvilinear searches. We then propose a general pattern search algorithmic framework that exploits curves of this type to carry out poll steps and for which we prove properties of asymptotic convergence to stationary points. We particularly point out that the proposed framework covers the case where search curves are arcs induced by the Euclidean projection of coordinate directions. The method is finally proved to arguably be superior, on smooth problems, than other recent projection-based algorithms and is competitive with state-of-the-art methods from the literature on constrained black-box optimization.

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arXiv2025

General Proximal Quasi-Newton Methods based on model functions for nonsmooth nonconvex problems

In this manuscript, we propose a general proximal quasi-Newton method tailored for nonconvex and nonsmooth optimization problems, where we do not require the sequence of the variable metric (or Hessian approximation) to be uniformly bounded as a prerequisite, instead, the variable metric is updated by a continuous matrix generator. From the respective of the algorithm, the objective function is approximated by the so-called local model function and subproblems aim to exploit the proximal point(s) of such model function, which help to achieve the sufficiently decreasing functional sequence along with the backtracking line search principle. Under mild assumptions in terms of the first-order information of the model function, every accumulation point of the generated sequence is stationary and the sequence of the variable metric is proved not to be bounded. Additionally, if the function has the Kurdyka-Łojasiewicz property at the corresponding accumulation point, we find that the whole sequence is convergent to the stationary point, and the sequence of the variable metric is proved to be uniformly bounded. Through the above results, we think that the boundedness of the sequence of the variable metric should depend on the regularity of objectives, rather than being assumed as a prior for nonsmooth optimization problems. Numerical experiments on polytope feasibility problems and (sparse) quadratic inverse problems demonstrate the effectiveness of our proposed model-based proximal quasi-Newton method, in comparison with the associated model-based proximal gradient method.

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arXiv2025

Convergence analysis of nonmonotone proximal gradient methods under local Lipschitz continuity and Kurdyka--Łojasiewicz property

The proximal gradient method is a standard approach for solving composite minimization problems in which the objective function is the sum of a continuously differentiable function and a lower semicontinuous, extended-valued function. The traditional convergence theory for both monotone and nonmonotone variants replies heavily on the assumption of global Lipschitz continuity of the gradient of the smooth part of the objective function. Recent work has shown that monotone proximal gradient methods converge globally only when the local (rather than global) Lipschitz continuity is assumed, provided that the Kurdyka--Łojasiewicz (KL) property holds. However, these results have not been extended to nonmonotone proximal gradient (NPG) methods. In this manuscript, we consider two types of NPG methods: those combined with the average line search and the max line search, respectively. By partitioning indices into two subsets, one of which aims to achieve a sufficient decrease in the functional sequence, we establish global convergence and rate-of-convergence results using the local Lipschitz continuity and the KL property, without requiring boundedness of the iterates. While finalizing this work, we noticed that [18] presented analogous results for the NPG method with average line search, but with a different partitioning strategy. Together, we confidently conclude that the convergence theory of the NPG method is independent on index partitioning choices.

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arXiv2023

Convergence Analysis of the Proximal Gradient Method in the Presence of the Kurdyka-Łojasiewicz Property without Global Lipschitz Assumptions

We consider a composite optimization problem where the sum of a continuously differentiable and a merely lower semicontinuous function has to be minimized. The proximal gradient algorithm is the classical method for solving such a problem numerically. The corresponding global convergence and local rate-of-convergence theory typically assumes, besides some technical conditions, that the smooth function has a globally Lipschitz continuous gradient and that the objective function satisfies the Kurdyka-Łojasiewicz property. Though this global Lipschitz assumption is satisfied in several applications where the objective function is, e.g., quadratic, this requirement is very restrictive in the non-quadratic case. Some recent contributions therefore try to overcome this global Lipschitz condition by replacing it with a local one, but, to the best of our knowledge, they still require some extra condition in order to obtain the desired global and rate-of-convergence results. The aim of this paper is to show that the local Lipschitz assumption together with the Kurdyka-Łojasiewicz property is sufficient to recover these convergence results.

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arXiv2023

Constrained composite optimization and augmented Lagrangian methods

We investigate finite-dimensional constrained structured optimization problems, featuring composite objective functions and set-membership constraints. Offering an expressive yet simple language, this problem class provides a modeling framework for a variety of applications. We study stationarity and regularity concepts, and propose a flexible augmented Lagrangian scheme. We provide a theoretical characterization of the algorithm and its asymptotic properties, deriving convergence results for fully nonconvex problems. It is demonstrated how the inner subproblems can be solved by off-the-shelf proximal methods, notwithstanding the possibility to adopt any solvers, insofar as they return approximate stationary points. Finally, we describe our matrix-free implementation of the proposed algorithm and test it numerically. Illustrative examples show the versatility of constrained composite programs as a modeling tool and expose difficulties arising in this vast problem class.

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arXiv2022

An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints

This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints are nonconvex and possibly complicated, but allow for a fast computation of projections onto this nonconvex set. Typical problem classes which satisfy this requirement are optimization problems with disjunctive constraints (like complementarity or cardinality constraints) as well as optimization problems over sets of matrices which have to satisfy additional rank constraints. The key idea behind our method is to keep these complicated constraints explicitly in the constraints and to penalize only the remaining constraints by an augmented Lagrangian function. The resulting subproblems are then solved with the aid of a problem-tailored nonmonotone projected gradient method. The corresponding convergence theory allows for an inexact solution of these subproblems. Nevertheless, the overall algorithm computes so-called Mordukhovich-stationary points of the original problem under a mild asymptotic regularity condition, which is generally weaker than most of the respective available problem-tailored constraint qualifications. Extensive numerical experiments addressing complementarity- and cardinality-constrained optimization problems as well as a semidefinite reformulation of MAXCUT problems visualize the power of our approach.

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