scholarly journals A Novel Approach for Solving Nonsmooth Optimization Problems with Application to Nonsmooth Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Hamid Reza Erfanian ◽  
M. H. Noori Skandari ◽  
A. V. Kamyad
Keyword(s):  
Nonsmooth Optimization ◽  
Optimization Problems ◽  
Piecewise Linear ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Piecewise Linear Function ◽  
Nonsmooth Equations ◽  
Smooth Functions ◽  
Generalized Derivative ◽  
Novel Approach

We present a new approach for solving nonsmooth optimization problems and a system of nonsmooth equations which is based on generalized derivative. For this purpose, we introduce the first order of generalized Taylor expansion of nonsmooth functions and replace it with smooth functions. In other words, nonsmooth function is approximated by a piecewise linear function based on generalized derivative. In the next step, we solve smooth linear optimization problem whose optimal solution is an approximate solution of main problem. Then, we apply the results for solving system of nonsmooth equations. Finally, for efficiency of our approach some numerical examples have been presented.

A comparative study of the PL homotopy and BFGS methods for some nonsmooth optimization problems

2019 ◽  
Vol 28 (2) ◽  
pp. 97-104
Author(s):  
ANDREI BOZANTAN ◽  
VASILE BERINDE
Keyword(s):  
Fixed Points ◽  
Nonsmooth Optimization ◽  
Optimization Problems ◽  
Piecewise Linear ◽  
Smooth Functions ◽  
Inexact Line Search ◽  
Homotopy Algorithm ◽  
Unconstrained Optimization Problems ◽  
Piecewise Linear Homotopy ◽  
Computation Of Fixed Points

We consider some non-smooth functions and investigate the numerical behavior of the Piecewise Linear Hompotopy (PLH) method ([Bozântan, A., An implementation of the piecewise-linear homotopy algorithm for the computation of fixed points, Creat. Math. Inform., 19 (2010), No.~2, 140–148] and [Bozântan, A. and Berinde, V., Applications of the PL homotopy algorithm for the computation of fixed points to unconstrained optimization problems, Creat. Math. Inform., 22 (2013), No. 1, 41–46]). We compare the PLH method with the BFGS with inexact line search, a quasi-Newton method, having some results reported in [Lewis, A. S. and Overton, M. L., Nonsmooth optimization via BFGS, submitted to SIAM J. Optimiz, (2009)]. For most of the considered cases, the characteristics of the PLH method are quite similar to the BFGS method, that is, the PLH method converges to local minimum values and the convergence rate seems to be linear with respect to the number of function evaluations, but we also identify some issues with the PLH method.

A Novel Approach to Designing Surrogate-assisted Genetic Algorithms by Combining Efficient Learning of Walsh Coefficients and Dependencies

2021 ◽  
Vol 1 (2) ◽  
pp. 1-23
Author(s):  
Arkadiy Dushatskiy ◽  
Tanja Alderliesten ◽  
Peter A. N. Bosman
Keyword(s):  
Boolean Functions ◽  
Surrogate Model ◽  
Optimization Problems ◽  
State Of The Art ◽  
Fitness Function ◽  
Optimal Solution ◽  
Problem Structure ◽  
Novel Approach ◽  
Efficient Learning ◽  
Nk Landscapes

Surrogate-assisted evolutionary algorithms have the potential to be of high value for real-world optimization problems when fitness evaluations are expensive, limiting the number of evaluations that can be performed. In this article, we consider the domain of pseudo-Boolean functions in a black-box setting. Moreover, instead of using a surrogate model as an approximation of a fitness function, we propose to precisely learn the coefficients of the Walsh decomposition of a fitness function and use the Walsh decomposition as a surrogate. If the coefficients are learned correctly, then the Walsh decomposition values perfectly match with the fitness function, and, thus, the optimal solution to the problem can be found by optimizing the surrogate without any additional evaluations of the original fitness function. It is known that the Walsh coefficients can be efficiently learned for pseudo-Boolean functions with k -bounded epistasis and known problem structure. We propose to learn dependencies between variables first and, therefore, substantially reduce the number of Walsh coefficients to be calculated. After the accurate Walsh decomposition is obtained, the surrogate model is optimized using GOMEA, which is considered to be a state-of-the-art binary optimization algorithm. We compare the proposed approach with standard GOMEA and two other Walsh decomposition-based algorithms. The benchmark functions in the experiments are well-known trap functions, NK-landscapes, MaxCut, and MAX3SAT problems. The experimental results demonstrate that the proposed approach is scalable at the supposed complexity of O (ℓ log ℓ) function evaluations when the number of subfunctions is O (ℓ) and all subfunctions are k -bounded, outperforming all considered algorithms.

scholarly journals Mixed integer nonlinear programming (MINLP)-based bandwidth utility function on internet pricing scheme with monitoring and marginal cost

2019 ◽  
Vol 9 (2) ◽  
pp. 1240
Author(s):  
Robinson Sitepu ◽  
Fitri Maya Puspita ◽  
Elika Kurniadi ◽  
Yunita Yunita ◽  
Shintya Apriliyani
Keyword(s):  
Utility Function ◽  
Marginal Cost ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Mixed Integer ◽  
Monitoring Cost ◽  
Internet Pricing ◽  
Pricing Scheme ◽  
Linear Optimization Problems

<span>The development of the internet in this era of globalization has increased fast. The need for internet becomes unlimited. Utility functions as one of measurements in internet usage, were usually associated with a level of satisfaction of users for the use of information services used. There are three internet pricing schemes used, that are flat fee, usage based and two-part tariff schemes by using one of the utility function which is Bandwidth Diminished with Increasing Bandwidth with monitoring cost and marginal cost. Internet pricing scheme will be solved by LINGO 13.0 in form of non-linear optimization problems to get optimal solution. The optimal solution is obtained using the either usage-based pricing scheme model or two-part tariff pricing scheme model for each services offered, if the comparison is with flat-fee pricing scheme. It is the best way for provider to offer network based on usage based scheme. The results show that by applying two part tariff scheme, the providers can maximize its revenue either for homogeneous or heterogeneous consumers.</span>

scholarly journals Hybrid Genetic Algorithm and Simulated Annealing for Function Optimization

2017 ◽  
Vol 1 (2) ◽  
pp. 82 ◽  
Author(s):  
Tirana Noor Fatyanosa ◽  
Andreas Nugroho Sihananto ◽  
Gusti Ahmad Fanshuri Alfarisy ◽  
M Shochibul Burhan ◽  
Wayan Firdaus Mahmudy
Keyword(s):  
Genetic Algorithm ◽  
Simulated Annealing ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Hybrid Genetic Algorithm ◽  
Optimal Solution ◽  
Function Optimization ◽  
Non Linear ◽  
Speed Up ◽  
Linear Problems

The optimization problems on real-world usually have non-linear characteristics. Solving non-linear problems is time-consuming, thus heuristic approaches usually are being used to speed up the solution’s searching. Among of the heuristic-based algorithms, Genetic Algorithm (GA) and Simulated Annealing (SA) are two among most popular. The GA is powerful to get a nearly optimal solution on the broad searching area while SA is useful to looking for a solution in the narrow searching area. This study is comparing performance between GA, SA, and three types of Hybrid GA-SA to solve some non-linear optimization cases. The study shows that Hybrid GA-SA can enhance GA and SA to provide a better result

Multiobjective Optimization of Multiloop Fractional Order PID Controller Tuned Using Bat Algorithm for Two Interacting Conical Tank Process

2014 ◽  
Vol 704 ◽  
pp. 373-379
Author(s):  
S.K. Lakshmanaprabu ◽  
U. Sabura Banu
Keyword(s):  
Fractional Order ◽  
Pid Controller ◽  
Controller Design ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Bat Algorithm ◽  
Absolute Error ◽  
Fractional Order Pid ◽  
Fractional Order Pid Controller

Multiloop fractional order PID controller is tuned using Bat algorithm for two interacting conical tank process. Two interacting conical tank process is modelled using mass balance equations. Two Interacting Conical Tank process is a complex system involving tedious interaction. Straight forward multiloop PID controller design involves various steps to design the controller. Due to easy implementation and quick convergence, Bat algorithm is used in recent past for solving continuous non-linear optimization problems to achieve global optimal solution. Bat algorithm, a swarm intelligence technique will be attempted to tune the multiloop fractional order PID controller for two interacting conical tank process. The multi objective optimized multiloop fractional PID controller is tested for tracking, disturbance rejection for minimum Integral time absolute error.

scholarly journals Security-Constrained Unit Commitment Based on a Realizable Energy Delivery Formulation

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Hongyu Wu ◽  
Qiaozhu Zhai ◽  
Xiaohong Guan ◽  
Feng Gao ◽  
Hongxing Ye
Keyword(s):  
Unit Commitment ◽  
Piecewise Linear ◽  
Optimal Solution ◽  
Piecewise Linear Function ◽  
Mixed Integer ◽  
Comparative Case Study ◽  
Dynamic Programming Method ◽  
Individual Unit ◽  
New Formulation ◽  
Bus System

Security-constrained unit commitment (SCUC) is an important tool for independent system operators in the day-ahead electric power market. A serious issue arises that the energy realizability of the staircase generation schedules obtained in traditional SCUC cannot be guaranteed. This paper focuses on addressing this issue, and the basic idea is to formulate the power output of thermal units as piecewise-linear function. All individual unit constraints and systemwide constraints are then reformulated. The new SCUC formulation is solved within the Lagrangian relaxation (LR) framework, in which a double dynamic programming method is developed to solve individual unit subproblems. Numerical testing is performed for a 6-bus system and an IEEE 118-bus system on Microsoft Visual C# .NET platform. It is shown that the energy realizability of generation schedules obtained from the new formulation is guaranteed. Comparative case study is conducted between LR and mixed integer linear programming (MILP) in solving the new formulation. Numerical results show that the near-optimal solution can be obtained efficiently by the proposed LR-based method.

scholarly journals Information-Maximization Clustering Based on Squared-Loss Mutual Information

2014 ◽  
Vol 26 (1) ◽  
pp. 84-131 ◽  
Author(s):  
Masashi Sugiyama ◽  
Gang Niu ◽  
Makoto Yamada ◽  
Manabu Kimura ◽  
Hirotaka Hachiya
Keyword(s):  
Mutual Information ◽  
Optimization Problems ◽  
Continuous Optimization ◽  
Optimal Solution ◽  
Selection Procedure ◽  
Eigenvalue Decomposition ◽  
Model Parameters ◽  
Computationally Efficient ◽  
Novel Approach ◽  
Information Maximization

Information-maximization clustering learns a probabilistic classifier in an unsupervised manner so that mutual information between feature vectors and cluster assignments is maximized. A notable advantage of this approach is that it involves only continuous optimization of model parameters, which is substantially simpler than discrete optimization of cluster assignments. However, existing methods still involve nonconvex optimization problems, and therefore finding a good local optimal solution is not straightforward in practice. In this letter, we propose an alternative information-maximization clustering method based on a squared-loss variant of mutual information. This novel approach gives a clustering solution analytically in a computationally efficient way via kernel eigenvalue decomposition. Furthermore, we provide a practical model selection procedure that allows us to objectively optimize tuning parameters included in the kernel function. Through experiments, we demonstrate the usefulness of the proposed approach.

SIMULATION OBJECTIVES OF FUZZY LINEAR PROGRAMMING WITH AN α-LEVEL METHOD OF λ-CONTINUE

2019 ◽  
Vol 6 (2) ◽  
pp. 71-76
Author(s):  
Alevtina Jur'evna Shatalova ◽  
Konstantin Andreevich Lebedev Konstantin Andreevich
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Investment Project ◽  
Fuzzy Linear Programming ◽  
Distribution Law ◽  
Linear Optimization Problems ◽  
Level Method

The article describes an approach that allows to formally describe the arising uncertainties in linear optimization problems. The generalized parametric alpha-level method of lambda-continuation of the fuzzy linear programming problem is considered. The model offers two methods that take into account the expansion of the binary fuzzy ratio (“strong” and “weak”). After the condition is formed taking into account the incoming quantities in the form of fuzzy numbers (the objective function and the system of constraints), the optimal solution (the value of the objective function) for each alpha and lambda is calculated using the simplex method implemented in Mathcad. On its basis, a mathematical model is built that will take into account the random values of alpha and lambda with a uniform distribution law. The paper presents a description of the simulation study, which confirms the conclusions about the possibilities of the method. Using the proposed theory, the decision-maker receives more information showing the behavior of the system with small changes in the input parameters to make more informed conclusions about the choice of financing of an investment project. The developed method of simulation of fuzzy estimation can be applied to other economic models with the appropriate necessary modification, for example, to assess the creditworthiness of the enterprise.

scholarly journals Leaders and followers algorithm for constrained non-linear optimization

2019 ◽  
Vol 13 (1) ◽  
pp. 162
Author(s):  
Helen Yuliana Angmalisang ◽  
Syaiful Anam ◽  
Sobri Abusini
Keyword(s):  
Constrained Optimization ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Linear Constraints ◽  
High Dimensional ◽  
Benchmark Problems ◽  
Leaders And Followers ◽  
Non Linear ◽  
Dimensional Optimization

<p>Leaders and Followers algorithm was a novel metaheuristics proposed by Yasser Gonzalez-Fernandez and Stephen Chen. In solving unconstrained optimization, it performed better exploration than other well-known metaheuristics, e.g. Genetic Algorithm, Particle Swarm Optimization and Differential Evolution. Therefore, it performed well in multi-modal problems. In this paper, Leaders and Followers was modified for constrained non-linear optimization. Several well-known benchmark problems for constrained optimization were used to evaluate the proposed algorithm. The result of the evaluation showed that the proposed algorithm consistently and successfully found the optimal solution of low dimensional constrained optimization problems and high dimensional optimization with high number of linear inequality constraint only. Moreover, the proposed algorithm had difficulty in solving high dimensional optimization problem with non-linear constraints and any problem which has more than one equality constraint. In the comparison with other metaheuristics, Leaders and Followers had better performance in overall benchmark problems.</p>

scholarly journals Efficiency of MOMA-plus Method to Solve Some Fully Fuzzy L-R Triangular Multiobjective Linear Programs

2018 ◽  
Vol 10 (2) ◽  
pp. 77 ◽  
Author(s):  
Abdoulaye Compaoré ◽  
Kounhinir Somé ◽  
Joseph Poda ◽  
Blaise Somé
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Linear Programs ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Linear Optimization Problem ◽  
Novel Approach ◽  
Linear Optimization Problems ◽  
Single Objective

In this paper, we propose a novel approach for solving some fully fuzzy L-R triangular multiobjective linear optimization programs using MOMA-plus method (Kounhinir, 2017). This approach is composed of two relevant steps such as the converting of the fully fuzzy L-R triangular multiobjective linear optimization problem into a deterministic multiobjective linear optimization and the applying of the adapting MOMA-plus method. The initial version of MOMA-plus method is designed for multiobjective deterministic optimization (Kounhinir, 2017) and having already been tested on the single-objective fuzzy programs (Abdoulaye, 2017). Our new method allow to find all of the Pareto optimal solutions of a fully fuzzy L-R triangular multiobjective linear optimization problems obtained after conversion. For highlighting the efficiency of our approach a didactic numerical example is dealt with and obtained solutions are compared to Total Objective Segregation Method proposed by Jayalakslmi and Pandia (Jayalakslmi 2014).

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