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dc-programming for neural network optimizations

JOURNAL OF GLOBAL OPTIMIZATION 2024年 1-17页

作者： Awasthi, Pranjal Mao, Anqi Mohri, Mehryar Zhong, Yutao Google Res New York NY 10011 USA Courant Inst New York NY 10012 USA

We discuss two key problems related to learning and optimization of neural networks: the computation of the adversarial attack for adversarial robustness and approximate optimization of complex functions. We show that both problems can be cast as instances of dc-programming. We give an explicit decomposition of the corresponding functions as differences of convex functions (dc) and report the results of experiments demonstrating the effectiveness of the dcA algorithm applied to these problems.

关键词： Neural networks Adversarial robustness Function approximation Optimization dc-programming

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DETERMINISTIC GLOBAL OPTIMIZATION WITH PARTITION SETS WHOSE FEASIBILITY IS NOT KNOWN - APPLICATION TO CONCAVE MINIMIZATION, REVERSE CONVEX CONSTRAINTS, dc-programming, AND LIPSCHITZIAN OPTIMIZATION

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1988年第1期58卷 11-37页

作者： HORST, R 1.Fachbereich IV-Mathematik Universit?t Trier Trier West Germany

A crucial problem for many global optimization methods is how to handle partition sets whose feasibility is not known. This problem is solved for broad classes of feasible sets including convex sets, sets defined by finitely many convex and reverse convex constraints, and sets defined by Lipschitzian inequalities. Moreover, a fairly general theory of bounding is presented and applied to concave objective functions, to functions representable as differences of two convex functions, and to Lipschitzian functions. The resulting algorithms allow one to solve any global optimization problem whose objective function is of one of these forms and whose feasible set belongs to one of the above classes. In this way, several new fields of optimization are opened to the application of global methods.

关键词： Global optimization nonconvex programming mathematical programming concave minimization dc-programming Lipschitzian optimization branch-and-bound methods

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Combining dc-programming and steepest-descent to solve the single-vehicle inventory routing problem

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COMPUTERS & INDUSTRIAL ENGINEERING 2011年第2期61卷 313-321页

作者： Zhong, Yiqing Aghezzaf, El-Houssaine Univ Ghent Dept Ind Management B-9052 Zwijnaarde Belgium

The single-vehicle cyclic inventory routing problem (SV-CIRP) is concerned with a repeated distribution of a product from a single depot to a selected subset of retailers having stable demands. If a retailer is selected for replenishment, the supplier collects a retailer-related fixed reward. The objective is to determine the subset of retailers to cyclically replenish, the quantities to be delivered to each, and to design the vehicle delivery routes so that the expected total distribution and inventory cost is minimized while the total collected rewards from the selected retailers is maximized. The resulting distribution plan must prevent stockouts from occurring at each retailer. In this paper, the underlying optimization problem for the SV-CIRP is formulated as a mixed-integer program with linear constraints and a nonlinear objective function. An optimization approach combining dc-programming and Branch-and-Bound within a steepest descent hybrid algorithm, denoted by dcA-SDHA, is developed for its solution. The approach is tested on some randomly generated problems and the obtained results are compared with results from the standard steepest descent hybrid algorithm (SDHA). These encouraging results show that the proposed approach is indeed computationally more effective and is worth being further investigated for the solution of medium to large instances. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Inventory-routing Nonlinear mixed-integer programming dc-programming Global optimization

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A Combined dc programming and Branch-and-Bound to Solve the Single-Vehicle Cyclic Inventory Routing Problem with a Fixed Cycle Time

A Combined DC Programming and Branch-and-Bound to Solve the ...

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International Conference on Computers and Industrial Engineering (CIE39)

作者： Zhong, Yiqing Aghezzaf, El-Houssaine Univ Ghent Dept Ind Management B-9052 Ghent Belgium

ISBN: (纸本)9781424441358

The single-vehicle cyclic inventory routing problem (SV-CIRP) consists of distributing a given product, on a cyclical basis, from a single depot to a selected subset of retailers. A fixed retailer-related reward is collected if a retailer is selected for the cyclic replenishment. The objective is thus to select a subset of retailers to be replenished and to develop an optimal cyclic distribution plan that minimizes expected distribution and inventory costs and in the same time maximizes the sum of rewards collected from the selected retailers. This distribution plan must prevent stockout from occurring at any of the selected retailers during each cycle. The underlying optimization problem for this SV-CIRP can be formulated as a mixed-integer program with linear constraints and a nonlinear objective function. In this paper, a dc-programming approach combined with Branch-and-Bound is used to solve the SV-CIRP with a fixed cycle time. The approach is then tested on some cases given in the literature, and the obtained results are compared with the results from the usual Branch-and-Bound. These preliminary results show that dc-programming combined with Branch-and-Bound is more effective that the usual Branch-and-Bound based LP-relaxations. The approach is therefore worth being investigated further for the solution of the cyclic inventory routing problem (CIRP).

关键词： Inventory Transportation Nonlinear Mixed-Integer programming dc-programming

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DELETION-BY-INFEASIBILITY RULE FOR dc-CONSTRAINED GLOBAL OPTIMIZATION

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1995年第2期84卷 443-455页

作者： FULOP, J HUNGARIAN ACAD SCI INST COMP & AUTOMATOPERAT RES & DECIS SYST LABBUDAPESTHUNGARY

In Ref. 1, a general class of branch-and-bound methods was proposed by Horst for solving global optimization problems. One of the main contributions of Ref. 1 was the opportunity of handling partition elements whose feasibility is not known. Deletion-by-infeasibility rules were presented for problems where the feasible set is convex, is defined by finitely many convex and reverse convex constraints, or is defined by Lipschitzian inequalities. In this note, we propose a new deletion-by-infeasibility rule for problems whose feasible set is defined by functions representable as differences of convex functions.

关键词： GLOBAL OPTIMIZATION NONCONVEX programming MATHEMATICAL programming dc-programming BRANCH-AND-BOUND METHODS

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Energy-Efficient Power Allocation for NOMA With Imperfect CSI

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 2019年第1期68卷 1009-1013页

作者： Zamani, Mohammad Reza Eslami, Mohsen Khorramizadeh, Mostafa Ding, Zhiguo Shiraz Univ Technol Dept Elect & Elect Engn Shiraz *** Iran Shiraz Univ Technol Dept Math Shiraz *** Iran Univ Lancaster Sch Comp & Commun Lancaster LA1 4YW England

An optimum user power allocation solution to maximize the energy efficiency (EE) in a non-orthogonal multiple access (NOMA) system with imperfect channel state information at transmitter (CSIT) and user quality of service constraint is proposed. The EE objective function is a non-convex nonlinear function that leads to a fractional programming problem. To deal with this problem, an auxiliary variable is defined, which converts the problem into a convex one. A closed form optimal solution for the power allocated to each user is found using the Karush-Kuhn-Tucker conditions. Furthermore, difference of two concave functions-dc-programming and Dinkelbach structure are used to find close-to-optimal total power value (sum of powers being allocated to all users) for the multiuser NOMA system. Simulation results show that in terms of EE, for users with low rate requirement and with CSIT error, NOMA has a better performance compared with conventional orthogonal multiple access (OMA).

关键词： Non-orthogonal multiple access (NOMA) imperfect CSI energy efficiency Dinkelbach method dc-programming

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Bilevel programming problems with simple convex lower level

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OPTIMIZATION 2016年第6期65卷 1203-1227页

作者： Mehlitz, Patrick Tech Univ Bergakad Freiberg Fac Math & Comp Sci D-09596 Freiberg Germany

This article is dedicated to the study of bilevel optimal control problems equipped with a fully convex lower level of special structure. In order to construct necessary optimality conditions, we consider a general bilevel programming problem in Banach spaces possessing operator constraints, which is a generalization of the original bilevel optimal control problem. We derive necessary optimality conditions for the latter problem using the lower level optimal value function, ideas from dc-programming and partial penalization. Afterwards, we apply our results to the original optimal control problem to obtain necessary optimality conditions of Pontryagin-type. Along the way, we derive a handy formula, which might be used to compute the subdifferential of the optimal value function which corresponds to the lower level parametric optimal control problem.

关键词： Bilevel programming optimization in Banach spaces nonsmooth optimization dc-programming optimal control

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Signal and image approximation with level-set constraints

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COMPUTING 2007年第2-3期81卷 137-160页

作者： Schnoerr, C. Univ Heidelberg Dept Math & Comp Sci D-69120 Heidelberg Germany

We present a novel variational approach to signal and image approximation using filter statistics (histograms) as constraints. Given a set of linear filters, we study the problem to determine the closest point to given data while constraining the level-sets of the filter outputs. This criterion and the constraints are formulated as a bilevel optimization problem. We develop an algorithm by representing the lower-level problem through complementarity constraints and by applying an interior-penalty relaxation method. Based on a decomposition of the penalty term into the difference of two convex functions, the resulting algorithm approximates the data by solving a sequence of convex programs. Our approach allows to model and to study the generation of image structure through the interaction of two convex processes for spatial approximation and for preserving filter statistics, respectively.

关键词： level-sets image approximation equilibrium constraints complementarity constraints dc-programming

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Energy-Efficient Secrecy Precoding for General MIMOME Wiretap Channels

Energy-Efficient Secrecy Precoding for General MIMOME Wireta...

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7th IEEE International Conference on Wireless Communications and Signal Processing (WCSP)

作者： Fang, Bing Qian, Zuping Zhong, Wei Shao, Wei PLA Univ Sci & Technol Coll Commun Engn Nanjing Jiangsu Peoples R China

ISBN: (纸本)9781467376877

In this paper, we study the energy-efficient secrecy precoding problem for a general multiple-input multiple-output multiple-eavesdropper (MIMOME) wiretap channel. The system model considered here consists of a common transmitter, a legitimate receiver, and multiple eavesdroppers, all mounted with multiple antennas. In order to improve the energy efficiency (EE) performance of the proposed system, an artificial noise (AN)-aided precoding scheme is adopted. Because the formulated EE maximization problem is of huge nonconvex complexity, we solve it by employing a two-tier optimization structure. Although the outer-tier optimization problem is quite simple and can be efficiently solved by the famous golden section searching method, the inner-tier optimization problem turns out to be nonconvex and cannot be easily solved. To solve it, we relax it to a secrecy rate maximization (SRM) problem, which naturally constitutes a difference convex (dc)-type programming problem and can be iteratively solved by employing a successive convex approximation (SCA) method. Numerical simulations are further provided to demonstrate the proposed algorithms. Results show that our algorithms can converge fast to a near-optimal solution.

关键词： MIMOME wiretap channel physical layer security energy efficiency dc-programming convex optimization

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Signal and image approximation with level-set constraints

Signal and image approximation with level-set constraints

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Conference on Variational and PDE Level Set Methods

作者： Schnoerr, C. Univ Heidelberg Dept Math & Comp Sci D-69120 Heidelberg Germany

关键词： level-sets image approximation equilibrium constraints complementarity constraints dc-programming

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