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On local search in d.c. optimization problems

APPLIED MATHEMATICS AND COMPUTATION 2015年 255卷 73-83页

作者： Strekalovsky, Alexander S. Russian Acad Sci Siberian Branch Inst Syst Dynam & Control Theory Irkutsk 664033 Russia

First, we consider a d.c. minimization problem with a simple feasible set and develop a special method based on the linearization with respect to the basic nonconvexity. The convergence of the methods is analyzed and compared with published results. Theoretical and practical stopping criteria are proposed. Second, we consider a problem with d.c. constraint and study the properties of special local search method for this problem. Finally, we consider a variant of local search for a general d.c. optimization problem and investigate its convergence. (C) 2014 Elsevier Inc. All rights reserved.

关键词： difference of two convex functions Local search Linearized problems Critical points

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THE DESIGN CENTERING PROBLEM AS A DC PROGRAMMING PROBLEM

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MATHEMATICAL PROGRAMMING 1988年第2期41卷 229-248页

作者： THACH, PT 1.Institute of Mathematics P.O. Box 631 Bo Ho Hanoi Vietnam

The following problem is studied: Given a compact setS inR n and a Minkowski functionalp(x), find the largest positive numberr for which there existsx ∈ S such that the set of ally ∈ R n satisfyingp(y?x) ≤ r is... 详细信息

The following problem is studied: Given a compact setS inR ⁿ and a Minkowski functionalp(x), find the largest positive numberr for which there existsx ∈ S such that the set of ally ∈ R ⁿ satisfyingp(y?x) ≤ r is contained inS. It is shown that whenS is the intersection of a closed convex set and several complementary convex sets (sets whose complements are open convex) this “design centering problem” can be reformulated as the minimization of some d.c. function (difference of two convex functions) overR ⁿ. In the case where, moreover,p(x) = (x ^T Ax)^1/2, withA being a symmetric positive definite matrix, a solution method is developed which is based on the reduction of the problem to the global minimization of a concave function over a compact convex set.

关键词： Design centering problem d.c. programming difference of two convex functions complementary convex sets reverse convex constraints global minimization of concave functions outer approximation algorithm

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General constructive representations for continuous piecewise-linear functions

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-REGULAR PAPERS 2004年第9期51卷 1889-1896页

作者： Wang, SN Tsing Hua Univ Dept Automat Beijing 100084 Peoples R China

The problem of constructing a canonical representation for an arbitrary continuous piecewise-linear (PWL) function in any dimension is considered in this paper. We solve the problem based on a general lattice PWL representation, which can be determined for a given continuous PWL function using existing methods. We first transform the lattice PWL representation into the difference of two convex functions, then propose a constructive procedure to rewrite the latter as a canonical representation that consists of at most n-level nestings of absolute-value functions in n dimensions, hence give a thorough solution to the problem mentioned above. In addition, we point out that there exist notable differences between a lattice representation and the two novel general constructive representations proposed in this paper, and explain that these differences make all the three representations be of their particular interests.

关键词： canonical representation difference of two convex functions general piecewise-linear (PWL) expression lattice PWL function

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Cluster oriented resource allocation and power optimisation for D2D network in cellular communications

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IET NEtwoRKS 2020年第4期9卷 170-179页

作者： Gour, Radhika Tyagi, Anshul Indian Inst Technol Roorkee Elect & Commun Engn Roorkee 247667 Uttarakhand India

This work proposes a cluster oriented channel assignment and difference of two convex functions (d.c.) programming based power optimisation algorithm for the downlink device-to-device (D2D) communication underlaying cellular networks. The authors' objective is to maximise D2D throughput while protecting the performance of existing cellular users (CUs) whose channel is reused among multiple D2D pairs, by imposing a minimum rate requirement constraint on each CU. The joint channel and power optimisation problem is a mixed integer non-linear programming problem that is NP-hard to solve. Therefore, a three stage solution is proposed: cluster formation to minimise the interference among the D2D pairs followed by an optimal channel assignment using the Hungarian algorithm and then an iterative power optimisation algorithm based on d.c. programming. Moreover, the transmission power of base station and D2D users is optimised on the same channel while considering the mutual interference among D2D pairs. Numerical results verify the effectiveness of the proposed resource allocation scheme in terms of the D2D sum rate and the number of successful transmission of D2D users. Moreover, the iterative power optimisation algorithm shows a fast convergence behaviour. In addition to this, the energy efficiency is also analysed with respect to various parameters.

关键词： nonlinear programming cellular radio integer programming resource allocation channel allocation convex programming iterative methods multiple D2D pairs minimum rate requirement constraint power optimisation problem mixed integer nonlinear programming problem cluster formation optimal channel assignment Hungarian algorithm iterative power optimisation algorithm transmission power base station cellular transmission resource allocation scheme D2D sum rate D2D maximum transmit power cellular communications cluster oriented channel assignment downlink device-to-device communication cellular networks cellular users difference of two convex functions dc programming based power optimisation algorithm

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Local search in problems with nonconvex constraints

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Computational Mathematics and Mathematical Physics 2007年第3期47卷 381-396页

作者： Gruzdeva, T.V. Strekalovsky, A.S. Institute of System Dynamics and Control Theory Siberian Division Russian Academy of Sciences Irkutsk 664033 ul. Lermontova 134 Russian Federation

Nonconvex optimization problems with an inequality constraint given by the difference of two convex functions (by a d.c. function) are considered. two methods for finding local solutions to this problem are proposed that combine the solution of partially linearized problems and descent to a level surface of the d.c. function. The convergence of the methods is analyzed, and stopping criterions are proposed. The methods are compared by testing them in a numerical experiment. © Nauka/Interperiodica 2007.

关键词： Critical point difference of two convex functions Level surface Linearized problem Local search

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A DC Programming Approach to Fractional Problems 11th

A DC Programming Approach to Fractional Problems

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11th International Conference on Learning and Intelligent Optimization (LION)

作者： Gruzdeva, Tatiana Strekalovsky, Alexander SB RAS Matrosov Inst Syst Dynam & Control Theory Lermontov Str 134 Irkutsk 664033 Russia

ISBN: (纸本)9783319694047;9783319694030

This paper addresses a rather general fractional optimization problem. There are two ways to reduce the original problem. The first one is a solution of an equation with the optimal value of an auxiliary d.c. optimization problem with a vector parameter. The second one is to solve the second auxiliary problem with nonlinear inequality constraints. Both auxiliary problems turn out to be d.c. optimization problems, which allows to apply Global Optimization Theory [1 1 ,12] and develop two corresponding global search algorithms that have been tested on a number of test problems from the recent publications.

关键词： Fractional optimization Nonconvex problem difference of two convex functions Equation with vector parameter Global search algorithm

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Power control for wireless cellular systems via DC programming

Power control for wireless cellular systems via DC programmi...

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14th IEEE/SP Workshop on Statistical Signal Processing

作者： Phan, Khoa T. Vorobyov, Sergiy A. Telambura, Chintha Le-Ngoc, Tho Univ Alberta Dept Elect & Comp Engn Edmonton AB Canada McGill Univ Dept Elect & Comp Engn Montreal PQ Canada

ISBN: (纸本)9781424411979

Power control at the base station is typically used in wireless cellular networks in order to optimize the transmission subject to quality of service (QoS) constraints. It has been shown that the power control problem in the wireless cellular network framework can be efficiently solved using the so-called geometric programming. However, in order to enable the application of geometric programmin the signal to interference ratio (SIR) has been considered instead of signal to interference plus noise ratio (SINR). Such problem reformulation is imprecise and might be loose because it does not take into account the noise component, especially for low signal to noise ratio (SNR) operation. In this paper, we show that the power control problem for wireless cellular systems can be efficiently solved via the so-called difference of two convex functions (D.C.) programming. Numerical simulation example demonstrates significant performance advantages of the proposed approach.

关键词： wireless communications cellular networks quality of service (QoS) difference of two convex functions DC programming

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