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SMOOTHED FUNCTIONALS AND CONSTRAINED STOCHASTIC-APPROXIMATION

SIAM JOURNAL ON NUMERICAL ANALYSIS 1988年第2期25卷 470-487页

作者： KREIMER, J RUBINSTEIN, RY

The problem of finding an extremum for a nonconvex function under convex constraints is considered. The original nonconvex function is replaced by an auxiliary one, called a smoothed function, which possesses some nice properties. Operating with the smoothed function and the given convex constraints the global extremum of the original problem is found.

关键词： 90 60 Monte Carlo method multiextremal optimization nonconvex programming smoothing stochastic approximation stochastic optimization

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Evolutionary Global Optimization, Manifolds and Applications 1st ed. 2016

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丛书名： Studies in Systems, Decision and Control

2016年

作者： Hime Aguiar e Oliveira Junior

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A branch and bound algorithm for the global optimization of Hessian Lipschitz continuous functions

A branch and bound algorithm for the global optimization of ...

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8th International Tri-Annual Conference on Optimization - Techniques and Applications (ICOTA)

作者： Fowkes, Jaroslav M. Gould, Nicholas I. M. Farmer, Chris L. Univ Edinburgh Sch Math Edinburgh EH9 3JZ Midlothian Scotland Rutherford Appleton Lab Computat Sci & Engn Dept Chilton OX11 0QX Oxon England Univ Oxford Math Inst Oxford OX1 3LB England

We present a branch and bound algorithm for the global optimization of a twice differentiable nonconvex objective function with a Lipschitz continuous Hessian over a compact, convex set. The algorithm is based on applying cubic regularisation techniques to the objective function within an overlapping branch and bound algorithm for convex constrained global optimization. Unlike other branch and bound algorithms, lower bounds are obtained via nonconvex underestimators of the function. For a numerical example, we apply the proposed branch and bound algorithm to radial basis function approximations.

关键词： Global optimization Lipschitzian optimization Branch and bound nonconvex programming

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Aggregate Constraint Homotopy Method for Nonlinear programming on Unbounded Set

Aggregate Constraint Homotopy Method for Nonlinear Programmi...

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International Conference on Intelligent Structure and Vibration Control (ISVC 2011)

作者： Xiao Yu Xiong Huijuan Yan Zhigang E China Jiaotong Univ Sch Basic Sci Nanchang 330013 Jiangxi Peoples R China Huazhong Agr Univ Dept Sci Wuhan 430070 Peoples R China Dongbei Univ Finance & Econ Accountancy Coll Dalian 116025 Peoples R China

ISBN: (纸本)9783037850206

In [1], an aggregate constraint aggregate (ACH) method for nonconvex nonlinear programming problems was presented and global convergence result was obtained when the feasible set is bounded and satisfies a weak normal cone condition with some standard constraint qualifications. In this paper, without assuming the boundedness of feasible set, the global convergence of ACH method is proven under a suitable additional assumption.

关键词： nonlinear programming nonconvex programming unbounded set aggregate constraint homotopy method

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When Can nonconvex Optimization Problems Be Solved with Gradient Descent? A Few Case Studies

When Can Nonconvex Optimization Problems Be Solved with Grad...

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作者： Gilboa, Dar Columbia University

学位级别：Ph.D.

Gradient descent and related algorithms are ubiquitously used to solve optimization problems arising in machine learning and signal processing. In many cases, these problems are nonconvex yet such simple algorithms are still effective. In an attempt to better understand this phenomenon, we study a number of nonconvex problems, proving that they can be solved efficiently with gradient descent. We will consider complete, orthogonal dictionary learning, and present a geometric analysis allowing us to obtain efficient convergence rate for gradient descent that hold with high probability. We also show that similar geometric structure is present in other nonconvex problems such as generalized phase retrieval Turning next to neural networks, we will also calculate conditions on certain classes of networks under which signals and gradients propagate through the network in a stable manner during the initial stages of training. Initialization schemes derived using these calculations allow training recurrent networks on long sequence tasks, and in the case of networks with low precision activation functions they make explicit a tradeoff between the reduction in precision and the maximal depth of a model that can be trained with gradient descent. We finally consider manifold classification with a deep feed-forward neural network, for a particularly simple configuration of the manifolds. We provide an end-to-end analysis of the training process, proving that under certain conditions on the architectural hyperparameters of the network, it can successfully classify any point on the manifolds with high probability given a sufficient number of independent samples from the manifold, in a timely manner. Our analysis relates the depth and width of the network to its fitting capacity and statistical regularity respectively in early stages of training.

关键词： Computer science Neural networks (Computer science) Machine learning nonconvex programming

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An efficient beamforming design for multipair full- duplex relaying systems

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ICT EXPRESS 2017年第1期3卷 9-13页

作者： Kim, Hyeon Min Van-Dinh Nguyen Shin, Oh-Soon Soongsil Univ Dept ICMC Convergence Technol Seoul 06978 South Korea Soongsil Univ Sch Elect Engn Seoul 06978 South Korea

We consider a decode-and-forward full-duplex relaying system for multiple pairs of users. Our objective is to maximize the minimum achievable rate for all user pairs under the transmit power constraints. We propose an iterative algorithm to solve the nonconvex max-min problem. In particular, the original problem is converted into successive convex programs by using an inner approximation method such that each iteration involves only a simple convex quadratic program. We show that the proposed algorithm improves achievement of the objective iteratively while guaranteeing convergence. Simulation results demonstrate that the proposed algorithm provides higher rates than both half-duplex and full-duplex relaying based on zero-forcing do. (C) 2017 The Korean Institute of Communications Information Sciences. Publishing Services by Elsevier B. V. This is an open access article under the CC BY-NC-ND license.

关键词： Full-duplex relaying Multipair relaying nonconvex programming Self-interference Transmit beamforming

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An algorithm for optimizing network flow capacity under economies of scale

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Journal of Optimization Theory and Applications 1975年第5期15卷 565-586页

作者： Bansai, P.P. Jacobsen, S.E. Engineering Systems Department School of Engineering and Applied Science University of California Los Angeles California United States

The problem of optimally allocating a fixed budget to the various arcs of a single-source, single-sink network for the purpose of maximizing network flow capacity is considered. The initial vector of arc capacities is given, and the cost function, associated with each arc, for incrementing capacity is concave;therefore, the feasible region is nonconvex. The problem is approached by Benders' decomposition procedure, and a finite algorithm is developed for solving the nonconvex relaxed master problems. A numerical example of optimizing network flow capacity, under economies of scale, is included. © 1975 Plenum Publishing Corporation.

关键词： mathematical programming network flows network synthesis nonconvex programming operations research

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Linear Precoding Designs for Amplify-and-Forward Multiuser Two-Way Relay Systems

Linear Precoding Designs for Amplify-and-Forward Multiuser T...

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54th Annual IEEE Global Telecommunications Conference (GLOBECOM)

作者： Wang, Rui Tao, Meixia Shanghai Jiao Tong Univ Dept Elect Engn Shanghai 200240 Peoples R China

ISBN: (纸本)9781424492688

We investigate the linear precoding designs for multiuser two-way relay system (MU-TWRS) where a multi-antenna base-station (BS) communicates with multiple single-antenna mobile stations (MSs) via a multi-antenna relay station (RS). The amplify-and-forward (AF) relay protocol is employed. The design goal is to optimize the precodings at BS, RS or both so as to minimize the total mean-square error (MSE) of the uplink messages while maintaining the individual signal-to-interference-plus-noise ratio (SINR) requirement for each downlink signal. We show that the BS precoding design problem can be converted to a standard second order cone programming (SOCP), while the RS precoding is non-convex for which a local optimal solution is obtained using an iterative algorithm. A joint BS-RS precoding is also obtained by alternating optimization of BS precoding and RS precoding with guaranteed convergence. Numerical results show that RS-precoding is superior to BS-precoding. Furthermore, the joint BS-RS precoding can significantly outperform the two individual precoding schemes. The implementation issues including complexity and feedback overhead are also discussed.

关键词： BS-precoding Downlink Interference Joints Optimization RS-precoding Relays Signal to noise ratio Vectors amplify and forward communication amplify and forward protocol concave programming cone programming interference suppression iterative algorithm iterative methods linear codes linear precoding design mean square error method mean square error methods mobile stations multi-antenna base station multi-antenna relay station multifrequency antennas multiuser channels multiuser two-way relay system nonconvex programming optimization precoding protocols signal to interference plus noise ratio uplink messages

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A new reformulation-linearization technique for bilinear programming problems

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Journal of Global Optimization 1992年第4期2卷 379-410页

作者： Sherali, Hanif D. Alameddine, Amine Department of Industrial and Systems Engineering Virginia Polytechnic Institute and State University Blacksburg 24061-0118 VA United States

This paper is concerned with the development of an algorithm for general bilinear programming problems. Such problems find numerous applications in economics and game theory, location theory, nonlinear multi-commodity network flows, dynamic assignment and production, and various risk management problems. The proposed approach develops a new Reformulation-Linearization Technique (RLT) for this problem, and imbeds it within a provably convergent branch-and-bound algorithm. The method first reformulates the problem by constructing a set of nonnegative variable factors using the problem constraints, and suitably multiplies combinations of these factors with the original problem constraints to generate additional valid nonlinear constraints. The resulting nonlinear program is subsequently linearized by defining a new set of variables, one for each nonlinear term. This "RLT" process yields a linear programming problem whose optimal value provides a tight lower bound on the optimal value to the bilinear programming problem. Various implementation schemes and constraint generation procedures are investigated for the purpose of further tightening the resulting linearization. The lower bound thus produced theoretically dominates, and practically is far tighter, than that obtained by using convex envelopes over hyper-rectangles. In fact, for some special cases, this process is shown to yield an exact linear programming representation. For the associated branch-and-bound algorithm, various admissible branching schemes are discussed, including one in which branching is performed by partitioning the intervals for only one set of variables x or y, whichever are fewer in number. Computational experience is provided to demonstrate the viability of the algorithm. For a large number of test problems from the literature, the initial bounding linear program itself solves the underlying bilinear programming problem. © 1992 Kluwer Academic Publishers.

关键词： Bilinear programming branch-and-bound global optimization nonconvex programming reformulation-linearization technique

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On programming with absolute-value functions

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Journal of Optimization Theory and Applications 1975年第1-2期17卷 181-183页

作者： Hill Jr., T.W. Ravindran, A. AiResearch Manufacturing Company of Arizona Phoenix Arizona United States Purdue University West Lafayette Indiana United States

This paper considers some programming problems with absolute-value (objective) functions subject to linear constraints. Necessary and sufficient conditions for the existence of finite optimum solutions to these proble... 详细信息

关键词： absolute-value problems linear programming Mathematical programming nonconvex programming operations research optimality criteria

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