D.C. Optimization: Theory, Methods and Algorithms

Tụy, Hoàng

doi:10.1007/978-1-4615-2025-2_4

Cited by 159 publications

(102 citation statements)

References 125 publications

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“…The special structure of the objective function of Problem (3) will be exploited in order to design a deterministic global optimization algorithm that allows us to find an optimal solution to the problem. More precisely, we will show that H(x) in (3) belongs to the broad class of DC functions, [11,10,21]. This key property will allow us to solve Problem (3) by branch-andbound algorithms, as the one described in Section 4, since lower and upper bounds can easily be obtained for DC functions as soon as a DC decomposition is available.…”

Section: Example 1 Let Us Consider the Network N = (A E) Withmentioning

confidence: 94%

“…An interesting property of the class of DC functions is that it is closed under the most common operations in optimization, [3,10,11,21,22]. In particular, if h 1 , .…”

Section: Example 1 Let Us Consider the Network N = (A E) Withmentioning

confidence: 99%

“…Contrary to the planar 1-median problem, known to be convex, [7], the 1-median problem on networks with continuous demand poses nontrivial challenges: we have no longer a simple expression for the objective, and its optimization calls for the use of global-optimization techniques. In particular, it is shown that the objective function is DC, [10,11,21,22] i.e., it can be written as the difference of two convex functions, and thus its optimization can be addressed via branch-and-bound methods customized for DC functions, [1,4,5].…”

Section: Introductionmentioning

confidence: 99%

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Solving the median problem with continuous demand on a network

Blanquero

Carrizosa

2013

Comput Optim Appl

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Where to locate one or several facilities on a network so as to minimize the expected users-closest facility transportation cost is a problem well studied in the OR literature under the name of median problem.In the median problem users are usually identified with nodes of the network. In many situations, however, such assumption is unrealistic, since users should be better considered to be distributed also along the edges of the transportation network. In this paper we address the median problem with demand distributed along edges and nodes. This leads to a globaloptimization problem, which can be solved to optimality by means of a branch-and-bound with DC bounds. Our computational experience shows that the problem is solved in short time even for large instances.

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Section: Example 1 Let Us Consider the Network N = (A E) Withmentioning

confidence: 94%

“…An interesting property of the class of DC functions is that it is closed under the most common operations in optimization, [3,10,11,21,22]. In particular, if h 1 , .…”

Section: Example 1 Let Us Consider the Network N = (A E) Withmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Solving the median problem with continuous demand on a network

Blanquero

Carrizosa

2013

Comput Optim Appl

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“…a nonconvex problem which can be described in terms of d.c. functions (difference of convex functions) and d.c. sets (difference of convex sets) [37]. By the fact that any constraint set can be equivalently relaxed by a nonsmooth indicator function, general nonconvex optimization problems can be written in the following standard d.c. programming form min{f (x) = g(x) − h(x) | ∀x ∈ X },…”

Section: Problems and Motivationmentioning

confidence: 99%

“…A more general model is that g(x) can be an arbitrary function [37]. Clearly, this d.c. programming problem is artificial.…”

Section: Problems and Motivationmentioning

confidence: 99%

On modeling and global solutions for d.c. optimization problems by canonical duality theory

Gao

2017

Applied Mathematics and Computation

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This paper presents a canonical d.c. (difference of canonical and convex functions) programming problem, which can be used to model general global optimization problems in complex systems. It shows that by using the canonical duality theory, a large class of nonconvex minimization problems can be equivalently converted to a unified concave maximization problem over a convex domain, which can be solved easily under certain conditions. Additionally, a detailed proof for triality theory is provided, which can be used to identify local extremal solutions. Applications are illustrated and open problems are presented.

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On the secrecy rate balancing problem of two‐way multiple‐input multiple‐output relaying networks using rank‐1 beamforming strategy

Fazeli

Akhlaghi

2022

Trans Emerging Tel Tech

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The physical layer security of a two‐way relay network (TWRN) in the presence of an eavesdropper is considered, where two multi‐antenna nodes try to exchange messages via a multi‐antenna two‐way relay node. It is assumed that the transmission occurs in two hops. Throughout the first hop, the messages are concurrently transmitted at the same time from two transmitting nodes to the relay, while an eavesdropper tries to overhear the signals. Throughout the second hop, to prevent eavesdropping, the relay broadcasts a combination of the received signals in the null‐space of the eavesdropper's channel gain vector. The objective is to explore the proper power allocation policy at transmitting ends as well as an effective beamforming strategy at the relay to address the secrecy rate balancing (SRB) problem. The radio frequency multiple‐input multiple‐output (RF‐MIMO) architecture design is proposed in which the relay can merely process one data stream; hence, the relay beamforming matrix is of rank‐one. The underlying problem is non‐convex. To tackle this issue, the original problem is divided into two sub‐problems, where using semi‐definite relaxation (SDR) and sequential parametric cone approximation (SPCA) techniques, are converted into convex forms, where suboptimal solutions are derived. Numerical results demonstrate effectiveness of the proposed method compared to the existing works addressed in the literature and also show that our proposed solution has better performance even for the secrecy energy efficiency (SEE) metric.

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D.C. Optimization: Theory, Methods and Algorithms

Cited by 159 publications

References 125 publications

Solving the median problem with continuous demand on a network

Solving the median problem with continuous demand on a network

On modeling and global solutions for d.c. optimization problems by canonical duality theory

On the secrecy rate balancing problem of two‐way multiple‐input multiple‐output relaying networks using rank‐1 beamforming strategy

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