• Title/Summary/Keyword: Nonconvex optimization

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SOLUTIONS OF NONCONVEX QUADRATIC OPTIMIZATION PROBLEMS VIA DIAGONALIZATION

  • YU, MOONSOOK;KIM, SUNYOUNG
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.5 no.2
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    • pp.137-147
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    • 2001
  • Nonconvex Quadratic Optimization Problems (QOP) are solved approximately by SDP (semidefinite programming) relaxation and SOCP (second order cone programmming) relaxation. Nonconvex QOPs with special structures can be solved exactly by SDP and SOCP. We propose a method to formulate general nonconvex QOPs into the special form of the QOP, which can provide a way to find more accurate solutions. Numerical results are shown to illustrate advantages of the proposed method.

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ON DUALITY FOR NONCONVEX QUADRATIC OPTIMIZATION PROBLEMS

  • Kim, Moon-Hee
    • East Asian mathematical journal
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    • v.27 no.5
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    • pp.539-543
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    • 2011
  • In this paper, we consider an optimization problem which consists a nonconvex quadratic objective function and two nonconvex quadratic constraint functions. We formulate its dual problem with semidefinite constraints, and we establish weak and strong duality theorems which hold between these two problems. And we give an example to illustrate our duality results. It is worth while noticing that our weak and strong duality theorems hold without convexity assumptions.

An Improved Particle Swarm Optimization Adopting Chaotic Sequences for Nonconvex Economic Dispatch Problems (개선된 PSO 기법을 적용한 전력계통의 경제급전)

  • Jeong, Yun-Won;Park, Jong-Bae;Cho, Ki-Seon;Kim, Hyeong-Jung;Shin, Joong-Rin
    • The Transactions of The Korean Institute of Electrical Engineers
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    • v.56 no.6
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    • pp.1023-1030
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    • 2007
  • This paper presents a new and efficient approach for solving the economic dispatch (ED) problems with nonconvex cost functions using particle swarm optimization (PSO). Although the PSO is easy to implement and has been empirically shown to perform well on many optimization problems, it may easily get trapped in a local optimum when solving problems with multiple local optima and heavily constrained. This paper proposes an improved PSO, which combines the conventional PSO with chaotic sequences (CPSO). The chaotic sequences combined with the linearly decreasing inertia weights in PSO are devised to improve the global searching capability and escaping from local minimum. To verify the feasibility of the proposed method, numerical studies have been performed for two different nonconvex ED test systems and its results are compared with those of previous works. The proposed CPSO algorithm outperforms other state-of-the-art algorithms in solving ED problems, which consider valve-point and multi-fuels with valve-point effects.

An Improved Mean-Variance Optimization for Nonconvex Economic Dispatch Problems

  • Kim, Min Jeong;Song, Hyoung-Yong;Park, Jong-Bae;Roh, Jae-Hyung;Lee, Sang Un;Son, Sung-Yong
    • Journal of Electrical Engineering and Technology
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    • v.8 no.1
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    • pp.80-89
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    • 2013
  • This paper presents an efficient approach for solving economic dispatch (ED) problems with nonconvex cost functions using a 'Mean-Variance Optimization (MVO)' algorithm with Kuhn-Tucker condition and swap process. The aim of the ED problem, one of the most important activities in power system operation and planning, is to determine the optimal combination of power outputs of all generating units so as to meet the required load demand at minimum operating cost while satisfying system equality and inequality constraints. This paper applies Kuhn-Tucker condition and swap process to a MVO algorithm to improve a global minimum searching capability. The proposed MVO is applied to three different nonconvex ED problems with valve-point effects, prohibited operating zones, transmission network losses, and multi-fuels with valve-point effects. Additionally, it is applied to the large-scale power system of Korea. The results are compared with those of the state-of-the-art methods as well.

An Economic Dispatch Algorithm as Combinatorial Optimization Problems

  • Min, Kyung-Il;Lee, Su-Won;Moon, Young-Hyun
    • International Journal of Control, Automation, and Systems
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    • v.6 no.4
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    • pp.468-476
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    • 2008
  • This paper presents a novel approach to economic dispatch (ED) with nonconvex fuel cost function as combinatorial optimization problems (COP) while most of the conventional researches have been developed as function optimization problems (FOP). One nonconvex fuel cost function can be divided into several convex fuel cost functions, and each convex function can be regarded as a generation type (G-type). In that case, ED with nonconvex fuel cost function can be considered as COP finding the best case among all feasible combinations of G-types. In this paper, a genetic algorithm is applied to solve the COP, and the $\lambda$-P table method is used to calculate ED for the fitness function of GA. The $\lambda$-P table method is reviewed briefly and the GA procedure for COP is explained in detail. This paper deals with three kinds of ED problems, namely ED considering valve-point effects (EDVP), ED with multiple fuel units (EDMF), and ED with prohibited operating zones (EDPOZ). The proposed method is tested for all three ED problems, and the test results show an improvement in solution cost compared to the results obtained from conventional algorithms.

Economic Dispatch Algorithm as Combinatorial Optimization Problems (조합최적화문제로 접근한 경제급전 알고리즘 개발)

  • Min, Kyung-Il;Lee, Su-Won;Choi, In-Kyu;Moon, Young-Hyun
    • The Transactions of The Korean Institute of Electrical Engineers
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    • v.58 no.8
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    • pp.1485-1495
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    • 2009
  • This paper presents a novel approach to economic dispatch (ED) with nonconvex fuel cost function as combinatorial optimization problems (COP) while most of the conventional researches have been developed as function optimization problems (FOP). One nonconvex fuel cost function can be divided into several convex fuel cost functions, and each convex function can be regarded as a generation type (G-type). In that case, ED with nonconvex fuel cost function can be considered as COP finding the best case among all feasible combinations of G-types. In this paper, a genetic algorithm is applied to solve the COP, and the ${\lambda}-P$ function method is used to calculate ED for the fitness function of GA. The ${\lambda}-P$ function method is reviewed briefly and the GA procedure for COP is explained in detail. This paper deals with two kinds of ED problems, namely ED with multiple fuel units (EDMF) and ED with prohibited operating zones (EDPOZ). The proposed method is tested for all the ED problems, and the test results show an improvement in solution cost compared to the results obtained from conventional algorithms.

E. Coli Bacterial Foraging Optimization based Distribution Nonconvex Economic Dispatch Algorithm (E. Coli Bacterial Foraging 최적화에 기준한 배전용 Nonconvex 경제급전 알고리즘)

  • Lee, S.S.;Kim, D.H.;Kim, M.K.;Norbekov, N.;Lee, H.C.;Lee, S.K.;Park, J.K.;Moon, S.I.;Yoon, Y.T.
    • Proceedings of the KIEE Conference
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    • 2007.07a
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    • pp.569-570
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    • 2007
  • 본 논문에서는 대장균 먹이찾기(E. Coli Bacterial Foraging)를 최적화 기법에 도입한 사례로서 이를 이용하여 Nonconvex 배전용 경제급전 알고리즘(distribution economic dispatch: DED을 제안한다. 제안된 DED 알고리즘은 향후 지역이나 구역 발전사업자를 중심으로 배전계통운용시스템에 필요한 경제급전 알고리즘으로 활용 할 수 있다.

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Control System Synthesis Using BMI: Control Synthesis Applications

  • Chung, Tae-Jin;Oh, Hak-Joon;Chung, Chan-Soo
    • International Journal of Control, Automation, and Systems
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    • v.1 no.2
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    • pp.184-193
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    • 2003
  • Biaffine Matrix Inequality (BMI) is known to provide the most general framework in control synthesis, but problems involving BMI's are very difficult to solve because nonconvex optimization should be solved. In the previous paper, we proposed a new solver for problems involving BMI's using Evolutionary Algorithms (EA). In this paper, we solve several control synthesis examples such as Reduced-order control, Simultaneous stabilization, Multi-objective control, $H_{\infty}$ optimal control, Maxed $H_2$ / $H_{\infty}$control design, and Robust $H_{\infty}$ control. Each of these problems is formulated as the standard BMI form, and solved by the proposed algorithm. The performance in each case is compared with those of conventional methods.

Large-Scale Phase Retrieval via Stochastic Reweighted Amplitude Flow

  • Xiao, Zhuolei;Zhang, Yerong;Yang, Jie
    • KSII Transactions on Internet and Information Systems (TIIS)
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    • v.14 no.11
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    • pp.4355-4371
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    • 2020
  • Phase retrieval, recovering a signal from phaseless measurements, is generally considered to be an NP-hard problem. This paper adopts an amplitude-based nonconvex optimization cost function to develop a new stochastic gradient algorithm, named stochastic reweighted phase retrieval (SRPR). SRPR is a stochastic gradient iteration algorithm, which runs in two stages: First, we use a truncated sample stochastic variance reduction algorithm to initialize the objective function. The second stage is the gradient refinement stage, which uses continuous updating of the amplitude-based stochastic weighted gradient algorithm to improve the initial estimate. Because of the stochastic method, each iteration of the two stages of SRPR involves only one equation. Therefore, SRPR is simple, scalable, and fast. Compared with the state-of-the-art phase retrieval algorithm, simulation results show that SRPR has a faster convergence speed and fewer magnitude-only measurements required to reconstruct the signal, under the real- or complex- cases.

On the Formulation and Optimal Solution of the Rate Control Problem in Wireless Mesh Networks

  • Le, Cong Loi;Hwang, Won-Joo
    • The Journal of Korean Institute of Communications and Information Sciences
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    • v.32 no.5B
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    • pp.295-303
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    • 2007
  • An algorithm is proposed to seek a local optimal solution of the network utility maximization problem in a wireless mesh network, where the architecture being considered is an infrastructure/backbone wireless mesh network. The objective is to achieve proportional fairness amongst the end-to-end flows in wireless mesh networks. In order to establish the communication constraints of the flow rates in the network utility maximization problem, we have presented necessary and sufficient conditions for the achievability of the flow rates. Since wireless mesh networks are generally considered as a type of ad hoc networks, similarly as in wireless multi-hop network, the network utility maximization problem in wireless mesh network is a nonlinear nonconvex programming problem. Besides, the gateway/bridge functionalities in mesh routers enable the integration of wireless mesh networks with various existing wireless networks. Thus, the rate optimization problem in wireless mesh networks is more complex than in wireless multi-hop networks.