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Convergence and complexity guarantees for a wide class of descent algorithms in nonconvex multi-objective optimization

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OPERATIONS RESEARCH LETTERS 2024年 54卷

作者： Lapucci, Matteo Univ Florence DINFO Via Santa Marta 3 I-50139 Florence Italy

We address conditions for global convergence and worst-case complexity bounds of descent algorithms in nonconvex multi-objective optimization. Specifically, we define the concept of steepest-descent-related directions. We consider iterative algorithms taking steps along such directions, selecting the stepsize according to a standard Armijo-type rule. We prove that methods fitting this framework automatically enjoy global convergence properties. Moreover, we show that a slightly stricter property, satisfied by most known algorithms, guarantees the same complexity bound of O(c-2) as the steepest descent method. (c) 2024 Elsevier B.V. All rights reserved.

关键词： Multi -objective optimization descent algorithms Global convergence Worst -case complexity Steepest -descent -related

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Group descent algorithms for nonconvex penalized linear and logistic regression models with grouped predictors

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STATISTICS AND COMPUTING 2015年第2期25卷 173-187页

作者： Breheny, Patrick Huang, Jian Univ Iowa Dept Biostat Iowa City IA 52242 USA Univ Iowa Dept Stat & Actuarial Sci Iowa City IA 52242 USA

Penalized regression is an attractive framework for variable selection problems. Often, variables possess a grouping structure, and the relevant selection problem is that of selecting groups, not individual variables. The group lasso has been proposed as a way of extending the ideas of the lasso to the problem of group selection. Nonconvex penalties such as SCAD and MCP have been proposed and shown to have several advantages over the lasso;these penalties may also be extended to the group selection problem, giving rise to group SCAD and group MCP methods. Here, we describe algorithms for fitting these models stably and efficiently. In addition, we present simulation results and real data examples comparing and contrasting the statistical properties of these methods.

关键词： Optimization Penalized regression Group lasso descent algorithms

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A descent ALGORITHM FOR THE OPTIMAL CONTROL OF ReLU NEURAL NETWORK INFORMED PDEs BASED ON APPROXIMATE DIRECTIONAL DERIVATIVES

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SIAM JOURNAL ON OPTIMIZATION 2024年第3期34卷 2314-2349页

作者： Dong, Guozhi Hintermueller, Michael Papafitsoros, Kostas Cent South Univ Sch Math & Stat HNP LAMA Changsha 410083 Peoples R China Humboldt Univ Inst Math D-10099 Berlin Germany Weierstrass Inst Appl Anal & Stochast WIAS D-10117 Berlin Germany Queen Mary Univ London Sch Math Sci London E1 4NS England

We propose and analyze a numerical algorithm for solving a class of optimal control problems for learning-informed semilinear partial differential equations (PDEs). Such PDEs contain constituents that are in principle unknown and are approximated by nonsmooth ReLU neural networks. We first show that direct smoothing of the ReLU network with the aim of using classical numerical solvers can have disadvantages, such as potentially introducing multiple solutions for the corresponding PDE. This motivates us to devise a numerical algorithm that treats directly the nonsmooth optimal control problem, by employing a descent algorithm inspired by a bundle-free method. Several numerical examples are provided and the efficiency of the algorithm is shown.

关键词： optimal control of nonsmooth partial differential equations data-driven models neural networks bundle-free methods descent algorithms

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Gradient based biobjective shape optimization to improve reliability and cost of ceramic components

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OPTIMIZATION AND ENGINEERING 2020年第4期21卷 1359-1387页

作者： Doganay, O. T. Gottschalk, H. Hahn, C. Klamroth, K. Schultes, J. Stiglmayr, M. Univ Wuppertal Gaussstr 20 D-42119 Wuppertal Germany

We consider the simultaneous optimization of the reliability and the cost of a ceramic component in a biobjective PDE constrained shape optimization problem. A probabilistic Weibull-type model is used to assess the probability of failure of the component under tensile load, while the cost is assumed to be proportional to the volume of the component. Two different gradient-based optimization methods are suggested and compared at 2D test cases. The numerical implementation is based on a first discretize then optimize strategy and benefits from efficient gradient computations using adjoint equations. The resulting approximations of the Pareto front nicely exhibit the trade-off between reliability and cost and give rise to innovative shapes that compromise between these conflicting objectives.

关键词： Biobjective shape optimization Shape gradients Probability of failure descent algorithms

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A note on robust descent in differentiable optimization

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OPERATIONS RESEARCH LETTERS 2017年第5期45卷 530-532页

作者： Dussault, Jean-Pierre Univ Sherbrooke Dept Informat Sherbrooke PQ J1K 2R1 Canada

In this note, we show how to alleviate the catastrophic cancellations that occur when comparing function values in trust-region algorithms. The main original contribution is to successfully adapt the line search strat... 详细信息

关键词： Unconstrained nonlinear optimization descent algorithms Numerical accuracy

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Design of code division multiple access filters based on sampled fiber Bragg grating by using global optimization algorithms

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OPTIMIZATION AND ENGINEERING 2014年第3期15卷 677-695页

作者： Ivorra, Benjamin Mohammadi, Bijan Ramos, Angel Manuel Univ Complutense Madrid Dept Matemat Aplicada E-28040 Madrid Spain Univ Montpellier 2 F-31057 Toulouse 01 France CERFACS F-31057 Toulouse 01 France

In this article, we focus on the design of code division multiple access filters (used in data transmission) composed of a particular optical fiber called sampled fiber Bragg grating (SFBG). More precisely, we consider an inverse problem that consists in determining the effective refractive index profile of an SFBG that produces a given reflected spectrum. In order to solve this problem, we use an original multi-layers semi-deterministic global optimization method based on the search of suitable initial conditions for a given optimization algorithm. The results obtained with our optimization algorithms are compared, in term of complexity and final design, with those given by an hybrid genetic algorithm (the method generally considered in the literature for designing SFBGs).

关键词： Global optimization Genetic algorithms descent algorithms Optical code division multiple access Sampled fiber Bragg grating design

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The effect of Laplacian filter in adaptive unsharp masking for infrared image enhancement

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INFRARED PHYSICS & TECHNOLOGY 2011年第5期54卷 427-438页

作者： Ilk, H. Gokhan Jane, Onur Ilk, Ozlem Ankara Univ Fac Engn Dept Elect Engn TR-06100 Ankara Turkey Middle E Tech Univ Fac Arts & Sci Dept Stat TR-06531 Ankara Turkey

Image processing, in particular image enhancement techniques have been the focal point of considerable research activity in the last decade. With the aid of an existing image enhancement technique, adaptive unsharp masking (AUM), we propose a novel kernel to be used in AUM filtering in order to enhance discontinuities which occur on the edges of targets of interest in infrared (IR) images. The proposed method uses an adaptive filter approach where an objective function is minimized by using descent algorithms. The output IR image has better sharpness and contrast adjustment for the detection of targets in terms of objective quality metrics. Hence, the proposed method ensures that the edges of the targets in IR images are sharper and that the quality of contrast adjustment has its optimum level in terms of peak signal-to-noise ratios. (C) 2011 Elsevier B.V. All rights reserved.

关键词： Infrared image processing Optimum Wiener filter descent algorithms Peak signal-to-noise ratio Quasi-Newton methods Adaptive filter

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A linearly convergent derivative-free descent method for the second-order cone complementarity problem

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OPTIMIZATION 2010年第8期59卷 1173-1197页

作者： Pan, Shaohua Chen, Jein-Shan Natl Taiwan Normal Univ Dept Math Taipei 11677 Taiwan S China Univ Technol Sch Math Sci Guangzhou 510640 Peoples R China

We consider a class of derivative-free descent methods for solving the second-order cone complementarity problem (SOCCP). The algorithm is based on the Fischer-Burmeister (FB) unconstrained minimization reformulation of the SOCCP, and utilizes a convex combination of the negative partial gradients of the FB merit function FB as the search direction. We establish the global convergence results of the algorithm under monotonicity and the uniform Jordan P-property, and show that under strong monotonicity the merit function value sequence generated converges at a linear rate to zero. Particularly, the rate of convergence is dependent on the structure of second-order cones. Numerical comparisons are also made with the limited BFGS method used by Chen and Tseng (An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Program. 104(2005), pp. 293-327), which confirm the theoretical results and the effectiveness of the algorithm.

关键词： second-order cone complementarity problem Fischer-Burmeister function descent algorithms derivative-free methods linear convergence

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On minimizing some merit functions for nonlinear complementarity problems under H-differentiability

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2008年第1期139卷 127-140页

作者： Tawhid, M. A. Goffin, J. L. Thompson Rivers Univ Sch Adv Technol & Math Dept Math & Stat Kamloops BC V2C 5N3 Canada McGill Univ Fac Management Montreal PQ H3A 1G5 Canada

In this paper, we describe the H-differentials of some well known NCP functions and their merit functions. We show how, under appropriate conditions on an H-differential of f, minimizing a merit function corresponding to f leads to a solution of the nonlinear complementarity problem. Our results give a unified treatment of such results for C(1)-functions, semismooth-functions, and locally Lipschitzian functions. Illustrations are given to show the usefulness of our results. We present also a result on the global convergence of a derivative-free descent algorithm for solving the nonlinear complementarity problem.

关键词： H-differentiability semismooth-functions locally Lipschitzian functions nonlinear complementarity problems NCP functions merit function descent algorithms

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Analysis of the impact of nonlinear heat transfer laws on temperature distribution in irradiated biological tissues: Mathematical models and optimal controls

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JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS 2007年第2期13卷 217-254页

作者： Belmiloudi, A. INSA Rennes IRMAR F-35043 Rennes France

In this paper, we consider nonlinear control problems governed by some generalized transient bioheat transfer-type models with the nonlinear Robin boundary conditions. The control estimates the blood perfusion rate, the heat transfer parameter, the distributed energy source terms, and the heat flux due to the evaporation, which affect the effects of thermal physical properties on the transient temperature of biological tissues. The result can be very beneficial for thermal diagnostics in medical practices, for example, for laser surgery, photo and thermotherapy for regional hyperthermia often used in treatment of cancer. First, the mathematical models are introduced and the existence, uniqueness, and regularity of a solution of the state equation are proved as well as the stability and maximum principle under extra assumptions. Afterwards, the optimal control problem is formulated in order to control the online temperature given by radiometric measurement. We prove that an optimal solution exists and obtain necessary optimality conditions. Some strategy for numerical realization based on the adjoint variables are provided.

关键词： optimal criterion biological tissues nonlinear bioheat adjoint system nonlinear boundary control pointwise controllers parameter identification magnetic resonance imaging data assimilation descent algorithms

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