arXiv subjects

Xinmin Yang

Publications and source records attributed to Xinmin Yang.

36 records · Page 2Linked to original sources

arXiv2024

Improvements to steepest descent method for multi-objective optimization

In this paper, we propose a simple yet efficient strategy for improving the multi-objective steepest descent method proposed by Fliege and Svaiter (Math Methods Oper Res, 2000, 3: 479--494). The core idea behind this strategy involves incorporating a positive modification parameter into the iterative formulation of the multi-objective steepest descent algorithm in a multiplicative manner. This modification parameter captures certain second-order information associated with the objective functions. We provide two distinct methods for calculating this modification parameter, leading to the development of two improved multi-objective steepest descent algorithms tailored for solving multi-objective optimization problems. Under reasonable assumptions, we demonstrate the convergence of sequences generated by the first algorithm toward a critical point. Moreover, for strongly convex multi-objective optimization problems, we establish the linear convergence to Pareto optimality of the sequence of generated points. The performance of the new algorithms is empirically evaluated through a computational comparison on a set of multi-objective test instances. The numerical results underscore that the proposed algorithms consistently outperform the original multi-objective steepest descent algorithm.

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arXiv2023

The convergence rate of the accelerated proximal gradient algorithm for Multiobjective Optimization is faster than $O(1/k^2)$

In this paper, we propose a fast proximal gradient algorithm for multiobjective optimization, it is proved that the convergence rate of the accelerated algorithm for multiobjective optimization developed by Tanabe et al. can be improved from $O(1/k^2)$ to $o(1/k^2)$ by introducing different extrapolation term $\frac{k-1}{k+α-1}$ with $α>3$. Further, we establish the inexact version of the proposed algorithm when the error term is additive, which owns the same convergence rate. At last, the efficiency of the proposed algorithm is verified on some numerical experiments.

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arXiv2023

Global convergence of block proximal iteratively reweighted algorithm with extrapolation

In this paper, we propose a proximal iteratively reweighted algorithm with extrapolation based on block coordinate update aimed at solving a class of optimization problems which is the sum of a smooth possibly nonconvex loss function and a general nonconvex regularizer with a special structure. The proposed algorithm can be used to solve the $\ell_p(0<p<1)$ regularization problem by employing a updating strategy of the smoothing parameter. It is proved that there exists the nonzero extrapolation parameter such that the objective function is nonincreasing. Moreover, the global convergence and local convergence rate are obtained by using the Kurdyka-Łojasiewicz (KL) property on the objective function. Numerical experiments are given to indicate the efficiency of the proposed algorithm.

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arXiv2023

Learning to optimize by multi-gradient for multi-objective optimization

The development of artificial intelligence (AI) for science has led to the emergence of learning-based research paradigms, necessitating a compelling reevaluation of the design of multi-objective optimization (MOO) methods. The new generation MOO methods should be rooted in automated learning rather than manual design. In this paper, we introduce a new automatic learning paradigm for optimizing MOO problems, and propose a multi-gradient learning to optimize (ML2O) method, which automatically learns a generator (or mappings) from multiple gradients to update directions. As a learning-based method, ML2O acquires knowledge of local landscapes by leveraging information from the current step and incorporates global experience extracted from historical iteration trajectory data. By introducing a new guarding mechanism, we propose a guarded multi-gradient learning to optimize (GML2O) method, and prove that the iterative sequence generated by GML2O converges to a Pareto critical point. The experimental results demonstrate that our learned optimizer outperforms hand-designed competitors on training multi-task learning (MTL) neural network.

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arXiv2023

Barzilai-Borwein Descent Methods for Multiobjective Optimization Problems with Variable Trade-off Metrics

The imbalances and conditioning of the objective functions influence the performance of first-order methods for multiobjective optimization problems (MOPs). The latter is related to the metric selected in the direction-finding subproblems. Unlike single-objective optimization problems, capturing the curvature of all objective functions with a single Hessian matrix is impossible. On the other hand, second-order methods for MOPs use different metrics for objectives in direction-finding subproblems, leading to a high per-iteration cost. To balance per-iteration cost and better curvature exploration, we propose a Barzilai-Borwein descent method with variable metrics (BBDMO\_VM). In the direction-finding subproblems, we employ a variable metric to explore the curvature of all objectives. Subsequently, Barzilai-Borwein's method relative to the variable metric is applied to tune objectives, which mitigates the effect of imbalances. We investigate the convergence behaviour of the BBDMO\_VM, confirming fast linear convergence for well-conditioned problems relative to the variable metric. In particular, we establish linear convergence for problems that involve some linear objectives. These convergence results emphasize the importance of metric selection, motivating us to approximate the trade-off of Hessian matrices to better capture the geometry of the problem. Comparative numerical results confirm the efficiency of the proposed method, even when applied to large-scale and ill-conditioned problems.

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arXiv2023

On the Convergence of Newton-type Proximal Gradient Method for Multiobjective Optimization Problems

In a recent study, Ansary (Optim Methods Softw 38(3):570-590,2023) proposed a Newton-type proximal gradient method for nonlinear multiobjective optimization problems (NPGMO). However, the favorable convergence properties typically associated with Newton-type methods were not established for NPGMO in Ansary's work. In response to this gap, we develop a straightforward framework for analyzing the convergence behavior of the NPGMO. Specifically, under the assumption of strong convexity, we demonstrate that the NPGMO enjoys quadratic termination, superlinear convergence, and quadratic convergence for problems that are quadratic, twice continuously differentiable and twice Lipschitz continuously differentiable, respectively.

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arXiv2023

Barzilai-Borwein Proximal Gradient Methods for Multiobjective Composite Optimization Problems with Improved Linear Convergence

When minimizing a multiobjective optimization problem (MOP) using multiobjective gradient descent methods, the imbalances among objective functions often decelerate the convergence. In response to this challenge, we propose two types of the Barzilai-Borwein proximal gradient method for multi-objective composite optimization problems (BBPGMO). We establish convergence rates for BBPGMO, demonstrating that it achieves rates of $O(\frac{1}{\sqrt{k}})$, $O(\frac{1}{k})$, and $O(r^{k})(0<r<1)$ for non-convex, convex, and strongly convex problems, respectively. Furthermore, we show that BBPGMO exhibits linear convergence for MOPs with several linear objective functions. Interestingly, the linear convergence rate of BBPGMO surpasses the existing convergence rates of first-order methods for MOPs, which indicates its enhanced performance and its ability to effectively address imbalances from theoretical perspective. Finally, we provide numerical examples to illustrate the efficiency of the proposed method and verify the theoretical results.

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arXiv2023

Reference-Point-Based Branch and Bound Algorithm for Multiobjective Optimization

In this paper, a branch and bound algorithm that incorporates the decision maker's preference information is proposed for multiobjective optimization. In the proposed algorithm, a new discarding test is designed to check whether a box contains preferred solutions according to the preference information expressed by means of reference points. In this way, the proposed algorithm is able to gradually guide the search towards the region of interest on the Pareto fronts during the solution process. We prove that the proposed algorithm obtains $\varepsilon$-efficient solutions distributed in the region of interest. Moreover, lower bound on the total finite number of required iterations for predefined precision is also provided. Finally, the algorithm is illustrated with a number of test problems.

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arXiv2022

Variable Metric Method for Unconstrained Multiobjective Optimization Problems

In this paper, we propose a variable metric method for unconstrained multiobjective optimization problems (MOPs). First, a sequence of points is generated using different positive definite matrices in the generic framework. It is proved that accumulation points of the sequence are Pareto critical points. Then, without convexity assumption, strong convergence is established for the proposed method. Moreover, we use a common matrix to approximate the Hessian matrices of all objective functions, along which, a new nonmonotone line search technique is proposed to achieve a local superlinear convergence rate. Finally, several numerical results demonstrate the effectiveness of the proposed method.

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arXiv2022

A Modification Piecewise Convexification Method for Box-Constrained Non-Convex Optimization Programs

This paper presents a piecewise convexification method to approximate the whole approximate optimal solution set of non-convex optimization problems with box constraints. In the process of box division, we first classify the sub-boxes and only continue to divide only some sub-boxes in the subsequent division. At the same time, applying the $α$-based Branch-and-Bound ({\rm$α$BB}) method, we construct a series of piecewise convex relax sub-problems, which are collectively called the piecewise convexification problem of the original problem. Then, we define the (approximate) solution set of the piecewise convexification problem based on the classification result of sub-boxes. Subsequently, we derive that these sets can be used to approximate the global solution set with a predefined quality. Finally, a piecewise convexification algorithm with a new selection rule of sub-box for the division and two new termination tests is proposed. Several instances verify that these techniques are beneficial to improve the performance of the algorithm.

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arXiv2022

Convergence rates analysis of Interior Bregman Gradient Method for Vector Optimization Problems

In recent years, by using Bregman distance, the Lipschitz gradient continuity and strong convexity were lifted and replaced by relative smoothness and relative strong convexity. Under the mild assumptions, it was proved that gradient methods with Bregman regularity converge linearly for single-objective optimization problems (SOPs). In this paper, we extend the relative smoothness and relative strong convexity to vector-valued functions and analyze the convergence of an interior Bregman gradient method for vector optimization problems (VOPs). Specifically, the global convergence rates are $\mathcal{O}(\frac{1}{k})$ and $\mathcal{O}(r^{k})(0<r<1)$ for convex and relative strongly convex VOPs, respectively. Moreover, the proposed method converges linearly for VOPs that satisfy a vector Bregman-PL inequality.

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arXiv2022

Memory gradient method for multiobjective optimization

In this paper, we propose a new descent method, termed as multiobjective memory gradient method, for finding Pareto critical points of a multiobjective optimization problem. The main thought in this method is to select a combination of the current descent direction and past multi-step iterative information as a new search direction and to obtain a stepsize by virtue of two types of strategies. It is proved that the developed direction with suitable parameters always satisfies the sufficient descent condition at each iteration. Based on mild assumptions, we obtain the global convergence and the rates of convergence for our method. Computational experiments are given to demonstrate the effectiveness of the proposed method.

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arXiv2022

A Barzilai-Borwein Descent Method for Multiobjective Optimization Problems

The steepest descent method proposed by Fliege et al. motivates the research on descent methods for multiobjective optimization, which has received increasing attention in recent years. However, empirical results show that the Armijo line search often gives a very small stepsize along the steepest direction, which decelerates the convergence seriously. This paper points out that the issue is mainly due to the imbalances among objective functions. To address this issue, we propose a Barzilai-Borwein descent method for multiobjective optimization (BBDMO) that dynamically tunes gradient magnitudes using Barzilai-Borwein's rule in direction-finding subproblem. With monotone and nonmonotone line search techniques, it is proved that accumulation points generated by BBDMO are Pareto critical points, respectively. Furthermore, theoretical results indicate the Armijo line search can achieve a better stepsize in BBDMO. Finally, comparative results of numerical experiments are reported to illustrate the efficiency of BBDMO and verify the theoretical results.

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arXiv2022

General inertial smoothing proximal gradient algorithm for the relaxation of matrix rank minimization problem

We consider the exact continuous relaxation model of matrix rank minimization problem proposed by Yu and Zhang (Comput.Optim.Appl. 1-20, 2022). Motivated by the inertial techinique, we propose a general inertial smoothing proximal gradient algorithm(GIMSPG) for this kind of problems. It is shown that the singular values of any accumulation point have a common support set and the nonzero singular values have a unified lower bound. Besides, the zero singular values of the accumulation point can be achieved within finite iterations. Moreover, we prove that any accumulation point of the sequence generated by the GIMSPG algorithm is a lifted stationary point of the continuous relaxation model under the flexible parameter constraint. Finally, we carry out numerical experiments on random data and image data respectively to illustrate the efficiency of the GIMSPG algorithm.

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arXiv2022

Conditional gradient method for vector optimization

In this paper, we propose a conditional gradient method for solving constrained vector optimization problems with respect to a partial order induced by a closed, convex and pointed cone with nonempty interior. When the partial order under consideration is the one induced by the non-negative orthant, we regain the method for multiobjective optimization recently proposed by Assunção et al. (Comput Optim Appl 78(3):741--768, 2021). In our method, the construction of auxiliary subproblem is based on the well-known oriented distance function. Three different types of step size strategies (Armijio, adaptative and nonmonotone) are considered. Without any assumptions, we prove that stationarity of accumulation points of the sequences produced by the proposed method equipped with the Armijio or the nonmonotone step size rule. To obtain the convergence result of the method with the adaptative step size strategy, we introduce an useful cone convexity condition which allows to circumvent the intricate question of the Lipschitz continuity of Jocabian for the objective function. This condition helps us generalize the classical descent lemma to the vector optimization case. Under suitable convexity assumptions for the objective function, it is proved that all accumulation points of any generated sequences obtained by our method are weakly efficient solutions.

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arXiv2021

The Smoothing Proximal Gradient Algorithm with Extrapolation for the Relaxation of $l_0$ Regularization Problem

In this paper, we consider the exact continuous relaxation model of $l_0$ regularization problem which was given by Bian and Chen (SIAM J. Numer. Anal 58(1): 858-883, 2020) and propose a smoothing proximal gradient algorithm with extrapolation (SPGE) for this kind of problem. We show that any accumulation point of the sequence generated by SPGE algorithm is a lifted stationary point of the relaxation model for a fairly general choice of extrapolation parameter. Moreover, it is shown that the whole sequence generated by SPGE algorithm converges to a lifted stationary point of the relaxation model. The convergence rate of $o(\frac{1}{k^{1-σ}})(σ\in(0,1))$ with respect to squared proximal residual is established. Finally, we conduct three important numerical experiments to illustrate the faster efficiency of the SPGE algorithm compared with the smoothing proximal gradient(SPG) algorithm proposed by Bian and Chen (SIAM J. Numer. Anal 58(1):858-883, 2020).

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arXiv2021

A novel multiobjective evolutionary algorithm based on decomposition and multi-reference points strategy

Many real-world optimization problems such as engineering design can be eventually modeled as the corresponding multiobjective optimization problems (MOPs) which must be solved to obtain approximate Pareto optimal fronts. Multiobjective evolutionary algorithm based on decomposition (MOEA/D) has been regarded as a significantly promising approach for solving MOPs. Recent studies have shown that MOEA/D with uniform weight vectors is well-suited to MOPs with regular Pareto optimal fronts, but its performance in terms of diversity usually deteriorates when solving MOPs with irregular Pareto optimal fronts. In this way, the solution set obtained by the algorithm can not provide more reasonable choices for decision makers. In order to efficiently overcome this drawback, we propose an improved MOEA/D algorithm by virtue of the well-known Pascoletti-Serafini scalarization method and a new strategy of multi-reference points. Specifically, this strategy consists of the setting and adaptation of reference points generated by the techniques of equidistant partition and projection. For performance assessment, the proposed algorithm is compared with existing four state-of-the-art multiobjective evolutionary algorithms on benchmark test problems with various types of Pareto optimal fronts. According to the experimental results, the proposed algorithm exhibits better diversity performance than that of the other compared algorithms. Finally, our algorithm is applied to two real-world MOPs in engineering optimization successfully.

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arXiv2020

Risk Minimization, Regret Minimization and Progressive Hedging Algorithms

This paper begins with a study on the dual representations of risk and regret measures and their impact on modeling multistage decision making under uncertainty. A relationship between risk envelopes and regret envelopes is established by using the Lagrangian duality theory. Such a relationship opens a door to a decomposition scheme, called progressive hedging, for solving multistage risk minimization and regret minimization problems. In particular, the classical progressive hedging algorithm is modified in order to handle a new class of linkage constraints that arises from reformulations and other applications of risk and regret minimization problems. Numerical results are provided to show the efficiency of the progressive hedging algorithms.

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