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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 Fachbereich IV-Mathematik Universität 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： branch-and-bound methods concave minimization DC-programming Global optimization Lipschitzian optimization mathematical programming nonconvex programming

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Secure and Energy-Efficient Beamforming for Simultaneous Information and Energy Transfer

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 2017年第11期16卷 7523-7537页

作者： Nasir, Ali Arshad Hoang Duong Tuan Duong, Trung Q. Poor, H. Vincent King Fahd Univ Petr & Minerals Dept Elect Engn Dhahran Saudi Arabia Univ Technol Sydney Sch Elect & Data Engn Broadway NSW 2007 Australia Queens Univ Belfast Belfast BT7 1NN Antrim North Ireland Princeton Univ Dept Elect Engn Princeton NJ 08544 USA

Some next-generation wireless networks will likely involve the energy-efficient transfer of information and energy over the same wireless channel. Moreover, densification of such networks will make the physical layer more vulnerable to cyber attacks by potential multi-antenna eavesdroppers. To address these issues, this paper considers transmit time-switching (TS) mode, in which energy and information signals are transmitted separately in time by the base station (BS). This protocol is not only easy to implement but also delivers the opportunity for multi-purpose beamforming, in which energy beamformers can be used to jam eavesdroppers during wireless power transfer. In the presence of imperfect channel estimation and multi-antenna eavesdroppers, the energy and information beamformers and the transmit TS ratio are jointly optimized to maximize the worst-case user secrecy rate subject to energy constrained users' harvested energy thresholds and a BS transmit power budget. New robust path-following algorithms, which involve one simple convex quadratic program at each iteration are proposed for computational solutions of this difficult optimization problem and also the problem of secure energy efficiency maximization. The latter adds further complexity due to additional optimization variables appearing in the denominator of the secrecy rate function. Numerical results confirm that the performance of the proposed computational solutions is robust against channel uncertainties.

关键词： Secrecy rate secrecy energy efficiency wireless power transfer time switching beamforming nonconvex programming

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On a modified subgradient algorithm for dual problems via sharp augmented Lagrangian

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JOURNAL OF GLOBAL OPTIMIZATION 2006年第1期34卷 55-78页

作者： Burachik, RS Gasimov, RN Ismayilova, NA Kaya, CY Univ Fed Rio de Janeiro COPPE BR-21941972 Rio de Janeiro Brazil Osmangazi Univ Dept Ind Engn TR-26030 Bademlik Eskisehir Turkey Univ S Australia Sch Math & Stat Mawson Lakes SA 5095 Australia

We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condition for existence of a dual solution. Using a practical selection of the step-size parameters, we demonstrate the algorithm and its advantages on test problems, including an integer programming and an optimal control problem.

关键词： augmented Lagrangian nonconvex programming nonsmooth optimization sharp Lagrangian subgradient optimization

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CONCAVE MINIMIZATION VIA CONICAL PARTITIONS AND POLYHEDRAL OUTER APPROXIMATION

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MATHEMATICAL programming 1991年第2期50卷 259-274页

作者： HORST, R THOAI, NV BENSON, HP INST MATH HANOIVIETNAM UNIV FLORIDA GAINESVILLEFL 32611

An algorithm is proposed for globally minimizing a concave function over a compact convex set. This algorithm combines typical branch-and-bound elements like partitioning, bounding and deletion with suitably introduced cuts in such a way that the computationally most expensive subroutines of previous methods are avoided. In each step, essentially only few linear programming problems have to be solved. Some preliminary computational results are reported.

关键词： CONCAVE MINIMIZATION GLOBAL OPTIMIZATION NONLINEAR programming nonconvex programming BRANCH-AND-BOUND OUTER APPROXIMATION CUTTING PLANES

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A Deep-Cutting-Plane Technique for Reverse Convex Optimization

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IEEE TRANSACTIONS ON SYSTEMS MAN AND CYBERNETICS PART B-CYBERNETICS 2011年第4期41卷 1054-1060页

作者： Moshirvaziri, Khosrow Amouzegar, Mahyar A. Calif State Univ Long Beach Dept Informat Syst Long Beach CA 90840 USA RAND Corp Santa Monica CA 90401 USA

A large number of problems in engineering design and in many areas of social and physical sciences and technology lend themselves to particular instances of problems studied in this paper. Cutting-plane methods have traditionally been used as an effective tool in devising exact algorithms for solving convex and large-scale combinatorial optimization problems. Its utilization in nonconvex optimization has been also promising. A cutting plane, essentially a hyperplane defined by a linear inequality, can be used to effectively reduce the computational efforts in search of a global solution. Each cut is generated in order to eliminate a large portion of the search domain. Thus, a deep cut is intuitively superior in which it will exclude a larger set of extraneous points from consideration. This paper is concerned with the development of deep-cutting-plane techniques applied to reverse-convex programs. An upper bound and a lower bound for the optimal value are found, updated, and improved at each iteration. The algorithm terminates when the two bounds collapse or all the generated subdivisions have been fathomed. Finally, computational considerations and numerical results on a set of test problems are discussed. An illustrative example, walking through the steps of the algorithm and explaining the computational process, is presented.

关键词： Cutting-plane methods global optimization nonconvex programming reverse-convex programs (RCPs)

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A New Design Paradigm for Secure Full-Duplex Multiuser Systems

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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 2018年第7期36卷 1480-1498页

作者： Van-Dinh Nguyen Nguyen, Hieu V. Dobre, Octavia A. Shin, Oh-Soon Soongsil Univ Sch Elect Engn Seoul 06978 South Korea Soongsil Univ Dept ICMC Convergence Technol Seoul 06978 South Korea Mem Univ Fac Engn & Appl Sci St John NF A1B 3X5 Canada

We consider a full-duplex (FD) multiuser system where an FD base station (BS) is designed to simultaneously serve both downlink (DL) and uplink (UL) users in the presence of half-duplex eavesdroppers (Eves). The problem is to maximize the minimum (max-min) secrecy rate (SR) among all legitimate users, where the information signals at the FD-BS are accompanied with artificial noise to debilitate the Eves' channels. To enhance the max-min SR, a major part of the power budget should be allocated to serve the users with poor channel qualities, such as those far from the FD-BS, undermining the SR for other users, and thus compromising the SR per-user. In addition, the main obstacle in designing an FD system is due to the self-interference (SI) and co-channel interference (CCI) among users. We therefore propose an alternative solution, where the FD-BS uses a fraction of the time block to serve near DL users and far UL users, and the remaining fractional time to serve other users. The proposed scheme mitigates the harmful effects of SI, CCI, and multiuser interference, and provides system robustness. The SR optimization problem has a highly nonconcave and nonsmooth objective, subject to nonconvex constraints. For the case of perfect channel state information (CSI), we develop a low-complexity path-following algorithm, which involves only a simple convex program of moderate dimension at each iteration. We show that our path-following algorithm guarantees convergence at least to a local optimum. Then, we extend the path-following algorithm to the cases of partially known Eves' CSI, where only statistics of CSI for the Eves are known, and worst-case scenario in which Eves can employ a more advanced linear decoder. The merit of our proposed approach is further demonstrated by extensive numerical results.

关键词： Artificial noise full-duplex radios full-duplex self-interference fractional time allocation nonconvex programming transmit beamforming physical-layer security

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Solutions and optimality criteria for nonconvex quadratic-exponential minimization problem

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MATHEMATICAL METHODS OF OPERATIONS RESEARCH 2008年第3期67卷 479-491页

作者： Gao, David Yang Ruan, Ning Virginia Polytech Inst & State Univ Dept Math Blacksburg VA 24061 USA Shanghai Univ Sci & Technol Sch Management Shanghai 200093 Peoples R China

This paper presents a set of complete solutions and optimality conditions for a nonconvex quadratic-exponential optimization problem. By using the canonical duality theory developed by the first author, the nonconvex primal problem in n-dimensional space can be converted into an one-dimensional canonical dual problem with zero duality gap, which can be solved easily to obtain all dual solutions. Each dual solution leads to a primal solution. Both global and local extremality conditions of these primal solutions can be identified by the triality theory associated with the canonical duality theory. Several examples are illustrated.

关键词： duality theory nonconvex programming global optimization quadratic-exponential function nonlinear algebraic equation triality

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Secrecy Rate Beamforming for Multicell Networks With Information and Energy Harvesting

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2017年第3期65卷 677-689页

作者： Nasir, Ali Arshad Hoang Duong Tuan Duong, Trung Q. Poor, H. Vincent King Fahd Univ Petr & Minerals Dept Elect Engn Dhahran 31261 Saudi Arabia Univ Technol Sydney Fac Engn & Informat Technol Broadway NSW 2007 Australia Queens Univ Belfast Sch Elect Elect Engn & Comp Sci Belfast BT7 1NN Antrim North Ireland Princeton Univ Dept Elect Engn Princeton NJ 08544 USA

Considering a multicell network for the secure wireless information and power transfer, this paper studies the joint design of transmit beamformers at the base stations (BSs) and receive signal splitting ratios at the end users' equipment (UE). The primary concern in this work is the network internal security, where there may be a single multiantenna eavesdropper or there is a risk that any near user may accidentally eavesdrop on the received signal of any far user. The objective is to maximize the minimum secrecy user rate under BS transmit power and UE minimum harvested energy constraints. New path-following algorithms are proposed for computational solutions of these difficult nonconvex optimization problems. Each iteration involves one simple convex quadratic program. Numerical results confirm that the proposed algorithms converge quickly after few iterations having a low computational complexity.

关键词： Secure communication secrecy rate energy harvesting nonconvex programming wireless information and power transfer signal splitting transmit beamforming

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A Generalized Formulation for Group Selection via ADMM

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JOURNAL OF SCIENTIFIC COMPUTING 2024年第1期100卷 1-37页

作者： Ke, Chengyu Shin, Sunyoung Lou, Yifei Ahn, Miju Southern Methodist Univ Dept Operat Res & Engn Management Dallas TX 75205 USA Pohang Univ Sci & Technol Dept Math Pohang 37673 Gyeongbuk South Korea Univ North Carolina Chapel Hill Dept Math Chapel Hill NC 27599 USA Univ North Carolina Chapel Hill Sch Data Sci & Soc Chapel Hill NC 27599 USA

This paper studies a statistical learning model where the model coefficients have a pre-determined non-overlapping group sparsity structure. We consider a combination of a loss function and a regularizer to recover the desired group sparsity patterns, which can embrace many existing works. We analyze directional stationary solutions of the proposed formulation, obtaining a sufficient condition for a directional stationary solution to achieve optimality and establishing a bound of the distance from the solution to a reference point. We develop an efficient algorithm that adopts an alternating direction method of multiplier (ADMM), showing that the iterates converge to a directional stationary solution under certain conditions. In the numerical experiment, we implement the algorithm for generalized linear models with convex and nonconvex group regularizers to evaluate the model performance on various data types, noise levels, and sparsity settings.

关键词： ADMM Group sparsity Group variable selection nonconvex programming Directional stationary solutions

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An effective subgradient algorithm via Mifflin's line search for nonsmooth nonconvex multiobjective optimization

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2024年第2期319卷 505-516页

作者： Maleknia, Morteza Soleimani-damaneh, Majid Isfahan Univ Technol Dept Math Sci Esfahan *** Iran Univ Tehran Coll Sci Sch Math Stat & Comp Sci Tehran Iran

We propose a descent subgradient algorithm for unconstrained nonsmooth nonconvex multiobjective optimization problems. To find a descent direction, we present an iterative process that efficiently approximates the epsilon-subdifferential of each objective function. To this end, we develop a new variant of Mifflin's line search in which the subgradients are arbitrary and its finite convergence is proved under a semismooth assumption. To reduce the number of subgradient evaluations, we employ a backtracking line search that identifies the objectives requiring an improvement in the current approximation of the epsilon-subdifferential. Meanwhile, for the remaining objectives, new subgradients are not computed. Unlike bundle-type methods, the proposed approach can handle nonconvexity without the need for algorithmic adjustments. Moreover, the quadratic subproblems have a simple structure, and hence the method is easy to implement. We analyze the global convergence of the proposed method and prove that any accumulation point of the generated sequence satisfies a necessary Pareto optimality condition. Furthermore, our convergence analysis addresses a theoretical challenge in a recently developed subgradient method. Through numerical experiments, we observe the practical capability of the proposed method and evaluate its efficiency when applied to a diverse range of nonsmooth test problems.

关键词： Multiple objective programming Nonlinear optimization Nonsmooth optimization nonconvex programming Subgradient

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