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Multi-target elliptic positioning via difference of convex functions programming

SIGNAL PROCESSING 2025年 234卷

作者： Dang, Xudong Liu, Hongwei Yan, Junkun Xidian Univ Natl Key Lab Radar Signal Proc 2 South Taibai Rd Xian 710071 Shaanxi Peoples R China

Multi-target localization in a distributed multiple-input multiple-output radar is quite challenging as the correct measurement-target associations in each transmitter-receiver pair are unknown. In this paper, we address this difficult problem from a joint optimization perspective. The measurement-target association and multi- target localization are jointly formulated as an intractable mixed-integer optimization problem, which contains both discrete and continuous variables. We first develop an equivalent difference of convex functions (DC) representation for the non-convex Boolean constraint imposed on the association variables, making the problem tractable. Then, a DC algorithm is derived to efficiently solve the resulting optimization problem. Simulation results demonstrate that the proposed DC method is numerically accurate when compared to state-of-the-art methods.

关键词： Multi-target localization Data association difference of convex functions Distributed MIMO radar convex optimization

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Optimality conditions and DC-Dinkelbach-type algorithm for vector fractional programming problems with ratios of difference of convex functions

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OPTIMIZATION 2025年第2期74卷 427-457页

作者： Ghazi, A. Roubi, A. Hassan First Univ Fac Sci Tech Lab MISI Settat Morocco

\In this work, necessary optimality conditions of KKT type for (weak) Pareto optimality are derived and a DC-Dinkelbach-type algorithm is proposed for vector fractional mathematical programs with ratios of difference of convex (DC) functions, and DC constraints, by reducing the latter to a system of scalar parametric problems and using DC tools. The special case of vector fractional programs with ratios of convex functions is also analysed.

关键词： Vector programming fractional programming difference of convex functions optimality conditions Dinkelbach algorithm

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Data fitting with signomial programming compatible difference of convex functions

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OPTIMIZATION AND ENGINEERING 2023年第2期24卷 973-987页

作者： Karcher, Cody J. MIT Dept Aeronaut & Astronaut 77 Massachusetts Ave Cambridge MA 02139 USA

Signomial Programming (SP) has proven to be a powerful tool for engineering design optimization, striking a balance between the computational efficiency of Geometric Programming (GP) and the extensibility of more general methods for optimization. While techniques exist for fitting GP compatible models to data, no models have been proposed that take advantage of the increased modeling flexibility available in SP. Here, a new difference of Softmax Affine function is constructed by utilizing existing methods of GP compatible fitting in difference of convex (DC) functions. This new function class is fit to data in log-log space and becomes either a signomial or a set of signomials upon inverse transformation. Examples presented here include simple test cases in 1D and 2D, and a fit to the performance data of the NACA 24xx family of airfoils. In each case, RMS error is driven to less than 1%.

关键词： Data fitting Signomial programming Log-log convexity difference of convex functions

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THE BOOSTED difference of convex functions ALGORITHM FOR NONSMOOTH functions

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SIAM JOURNAL ON OPTIMIZATION 2020年第1期30卷 980-1006页

作者： Aragon Artacho, Francisco J. Vuong, Phan T. Univ Alicante Dept Math Alicante Spain Univ Southampton Sch Math Sci Southampton SO17 1BJ Hants England

The boosted difference of convex functions algorithm (BDCA) was recently proposed for minimizing smooth difference of convex (DC) functions. BDCA accelerates the convergence of the classical difference of convex functions algorithm (DCA) thanks to an additional line search step. The purpose of this paper is twofold. First, we show that this scheme can be generalized and successfully applied to certain types of nonsmooth DC functions, namely, those that can be expressed as the difference of a smooth function and a possibly nonsmooth one. Second, we show that there is complete freedom in the choice of the trial step size for the line search, which is something that can further improve its performance. We prove that any limit point of the BDCA iterative sequence is a critical point of the problem under consideration and that the corresponding objective value is monotonically decreasing and convergent. The global convergence and convergence rate of the iterations are obtained under the Kurdyka-Lojasiewicz property. Applications and numerical experiments for two problems in data science are presented, demonstrating that BDCA outperforms DCA. Specifically, for the minimum sum-of-squares clustering problem, BDCA was on average 16 times faster than DCA, and for the multidimensional scaling problem, BDCA was 3 times faster than DCA.

关键词： difference of convex functions boosted difference of convex functions algorithm Kurdyka-Lojasiewicz property clustering problem multidimensional scaling problem

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Optimality Conditions and a Method of Centers for Minimax Fractional Programs with difference of convex functions

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2020年第1期187卷 105-132页

作者： Boufi, Karima El Haffari, Mostafa Roubi, Ahmed Univ Hassan 1 Fac Sci & Tech Lab MISI Settat Morocco Univ Abdelmalek Essaadi Lab MISI Tetouan Morocco Univ Abdelmalek Essaadi Ecole Normale Super Tetouan Morocco

We are concerned in this paper with minimax fractional programs whose objective functions are the maximum of finite ratios of difference of convex functions, with constraints also described by difference of convex functions. Like Dinkelbach-type algorithms, the method of centers for generalized fractional programs fails to work for such problems, since the parametric subproblems may be nonconvex, whereas the latters need a global optimal solution for these subproblems. We first give necessary optimality conditions for these problems, by means of convex analysis tools, and then extend the last method to solve such programs. The method is based on solving a sequence of parametric convex problems. We show that every cluster point of the sequence of optimal solutions of these subproblems satisfies necessary optimality conditions of Karush-Kuhn-Tucker criticality type.

关键词： Fractional programming difference of convex functions Optimality conditions Method of centers

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Alternating direction method of multipliers with difference of convex functions

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2018年第3期44卷 723-744页

作者： Sun, Tao Yin, Penghang Cheng, Lizhi Jiang, Hao Natl Univ Def Technol Coll Sci Changsha 410073 Hunan Peoples R China Univ Calif Los Angeles Dept Math Los Angeles CA 90095 USA Natl Univ Def Technol Coll Sci Changsha 410073 Hunan Peoples R China Natl Univ Def Technol State Key Lab High Performance Computat Changsha 410073 Hunan Peoples R China Natl Univ Def Technol Coll Comp Changsha 410073 Hunan Peoples R China

In this paper, we consider the minimization of a class of nonconvex composite functions with difference of convex structure under linear constraints. While this kind of problems in theory can be solved by the celebrated alternating direction method of multipliers (ADMM), a direct application of ADMM often leads to difficult nonconvex subproblems. To address this issue, we propose to convexify the subproblems through a linearization technique as done in the difference of convex functions algorithm (DCA). By assuming the Kurdyka-Aojasiewicz property, we prove that the resulting algorithm sequentially converges to a critical point. It turns out that in the applications of signal and image processing such as compressed sensing and image denoising, the proposed algorithm usually enjoys closed-form solutions of the subproblems and thus can be very efficient. We provide numerical experiments to demonstrate the effectiveness of our algorithm.

关键词： Nonconvex Alternating direction method of multipliers difference of convex functions Kurdyka-Lojasiewicz property

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Global optimization method for maximizing the sum of difference of convex functions ratios over nonconvex region

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2013年第1-2期41卷 153-169页

作者： Pei, Yonggang Zhu, Detong Shanghai Normal Univ Math & Sci Coll Shanghai 200234 Peoples R China Shanghai Normal Univ Business Coll Shanghai 200234 Peoples R China

A global optimization algorithm is presented for maximizing the sum of difference of convex functions ratios problem over nonconvex feasible region. This algorithm is based on branch and bound framework. To obtain a difference of convex programming, the considered problem is first reformulated by introducing new variables as few as possible. By using subgradient and convex envelope, the fundamental problem of estimating lower bound in the branch and bound algorithm is transformed into a relaxed linear programming problem which can be solved efficiently. Furthermore, the size of the relaxed linear programming problem does not change during the algorithm search. Lastly, the convergence of the algorithm is analyzed and the numerical results are reported.

关键词： Global optimization Sum of ratios difference of convex functions Branch and bound

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Nonconvex Optimization Problems for Maximum Hands-Off Control

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2025年第3期70卷 1905-1912页

作者： Ikeda, Takuya Univ Kitakyushu Fac Environm Engn Fukuoka 8080135 Japan

Maximum hands-off control is the optimal solution to the L(0 )optimal control problem. While convex approximation is typically used to relax this problem, it does not necessarily result in maximum hands-off control. Therefore, this study introduces a nonconvex approximation method and a class of nonconvex optimal control problems that are always equivalent to the maximum hands-off control problem. A computation method based on difference of convex functions optimization is then derived and numerically validated.

关键词： Optimal control Optimization Vectors Cost function Approximation methods convex functions Sparse approximation State estimation Standards Optimization methods difference of convex functions nonconvex approximation optimal control sparse control

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A global interior point method for nonconvex geometric programming

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OPTIMIZATION AND ENGINEERING 2024年第2期25卷 605-+页

作者： do Nascimento, Roberto Quirino Santos, Rubia Mara de Oliveira Maculan, Nelson Univ Fed Paraiba Dept Sci Comp Joao Pessoa Brazil Univ Fed Mato Grosso do Sul Dept Math Campo Grande Brazil Univ Fed Rio De Janeiro Coppe Sistemas Rio De Janeiro Brazil

The strategy presented in this paper differs significantly from existing approaches as we formulate the problem as a standard optimization problem of difference of convex functions. We have developed the necessary and sufficient conditions for global solutions in this standard form. The main challenge in the standard form arises from a constraint of the form g(t) = 1, where g is a convex function. We utilize the classical inequality between the weighted arithmetic and harmonic means to overcome this challenge. This enables us to express the optimality conditions as a convex geometric programming problem and employ a predictor-corrector primal-dual interior point method for its solution, with weights updated during the predictor phase. The interior point method solves the dual problem of geometric programming and obtains the primal solution through exponential transformation. We have implemented the algorithm in Fortran 90 and validated it using a set of test problems from the literature. The proposed method successfully solved all the test problems, and the computational results are presented alongside the tested problems and the corresponding solutions found.

关键词： Global optimization difference of convex functions Geometric programming Interior point method

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A global two-stage algorithm for non-convex penalized high-dimensional linear regression problems

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COMPUTATIONAL STATISTICS 2023年第2期38卷 871-898页

作者： Li, Peili Liu, Min Yu, Zhou East China Normal Univ KLATASDS MOE Sch Stat Shanghai 200062 Peoples R China Wuhan Univ Sch Math & Stat Wuhan 430072 Peoples R China

By the asymptotic oracle property, non-convex penalties represented by minimax concave penalty (MCP) and smoothly clipped absolute deviation (SCAD) have attracted much attentions in high-dimensional data analysis, and have been widely used in signal processing, image restoration, matrix estimation, etc. However, in view of their non-convex and non-smooth characteristics, they are computationally challenging. Almost all existing algorithms converge locally, and the proper selection of initial values is crucial. Therefore, in actual operation, they often combine a warm-starting technique to meet the rigid requirement that the initial value must be sufficiently close to the optimal solution of the corresponding problem. In this paper, based on the DC (difference of convex functions) property of MCP and SCAD penalties, we aim to design a global two-stage algorithm for the high-dimensional least squares linear regression problems. A key idea for making the proposed algorithm to be efficient is to use the primal dual active set with continuation (PDASC) method to solve the corresponding sub-problems. Theoretically, we not only prove the global convergence of the proposed algorithm, but also verify that the generated iterative sequence converges to a d-stationary point. In terms of computational performance, the abundant research of simulation and real data show that the algorithm in this paper is superior to the latest SSN method and the classic coordinate descent (CD) algorithm for solving non-convex penalized high-dimensional linear regression problems.

关键词： High-dimensional linear regression Global convergence Two-stage algorithm Primal dual active set with continuation algorithm difference of convex functions

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