scholarly journals An Improved Subgradiend Optimization Technique for Solving IPs with Lagrangean Relaxation

2013 ◽  
Vol 61 (2) ◽  
pp. 135-140
Author(s):  
M Babul Hasan ◽  
Md Toha
Keyword(s):  
Dual Problem ◽  
Optimization Problems ◽  
Optimization Technique ◽  
Concave Function ◽  
Optimization Method ◽  
Duality Gap ◽  
Dual Function ◽  
Subgradient Optimization ◽  
Step Size ◽  
Primal Problem

The objective of this paper is to improve the subgradient optimization method which is used to solve non-differentiable optimization problems in the Lagrangian dual problem. One of the main drawbacks of the subgradient method is the tuning process to determine the sequence of step-lengths to update successive iterates. In this paper, we propose a modified subgradient optimization method with various step size rules to compute a tuning- free subgradient step-length that is geometrically motivated and algebraically deduced. It is well known that the dual function is a concave function over its domain (regardless of the structure of the cost and constraints of the primal problem), but not necessarily differentiable. We solve the dual problem whenever it is easier to solve than the primal problem with no duality gap. However, even if there is a duality gap the solution of the dual problem provides a lower bound to the primal optimum that can be useful in combinatorial optimization. Numerical examples are illustrated to demonstrate the method. DOI: http://dx.doi.org/10.3329/dujs.v61i2.17059 Dhaka Univ. J. Sci. 61(2): 135-140, 2013 (July)

scholarly journals The Use of the Duality Principle to Solve Optimization Problems

2018 ◽  
Vol 6 (1) ◽  
pp. 33
Author(s):  
Rowland Jerry Okechukwu Ekeocha ◽  
Chukwunedum Uzor ◽  
Clement Anetor
Keyword(s):  
Optimization Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Linear Program ◽  
Duality Gap ◽  
Duality Principle ◽  
Primal Problem ◽  
Optimal Values ◽  
Primal And Dual Problems ◽  
Primal And Dual

<p><span>The duality principle provides that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem. However the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition.<span>  </span>In other words given any linear program, there is another related linear program called the dual. In this paper, an understanding of the dual linear program will be developed. This understanding will give important insights into the algorithm and solution of optimization problem in linear programming. <span> </span>Thus the main concepts of duality will be explored by the solution of simple optimization problem.</span></p>

Design of Optimal CMOS Analog Amplifier Circuits Using a Hybrid Evolutionary Optimization Technique

2017 ◽  
Vol 27 (02) ◽  
pp. 1850029 ◽  
Author(s):  
Bishnu Prasad De ◽  
Kanchan Baran Maji ◽  
Rajib Kar ◽  
Durbadal Mandal ◽  
Sakti Prasad Ghoshal
Keyword(s):  
Power Dissipation ◽  
Optimization Problems ◽  
Optimization Technique ◽  
Optimization Method ◽  
Vlsi Circuits ◽  
Optimal Designs ◽  
Efficient Design ◽  
Mos Transistors ◽  
Evolutionary Technique ◽  
Random Particle

This paper proposes an efficient design technique for two commonly used VLSI circuits, namely, CMOS current mirror load-based differential amplifier circuit and CMOS two-stage operational amplifier. The hybrid evolutionary method utilized for these optimal designs is random particle swarm optimization with differential evolution (RPSODE). Random PSO utilizes the weighted particles for monitoring the search directions. DE is a robust evolutionary technique. It has demonstrated an exclusive performance for the optimization problems which are continuous and global but suffers from the uncertainty issues. PSO is a robust optimization method but suffers from sub-optimality problem. This paper effectively hybridizes the random PSO and DE to remove the limitations related to both the techniques individually. In this paper, RPSODE is employed to optimize the sizes of the MOS transistors to reduce the overall area taken by the circuit while satisfying the design constraints. The results obtained from RPSODE technique are validated in SPICE environment. SPICE-based simulation results justify that RPSODE is a much better technique than other formerly reported methods for the designs of the above mentioned circuits in terms of MOS area, gain, power dissipation, etc.

scholarly journals Extended Duality in Fuzzy Optimization Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Tingting Zou
Keyword(s):  
Nonlinear Programming ◽  
Original Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Fuzzy Optimization ◽  
Duality Gap ◽  
Continuous Variables ◽  
Objective Functions ◽  
Duality Theorems ◽  
Fuzzy Nonlinear Programming

Duality theorem is an attractive approach for solving fuzzy optimization problems. However, the duality gap is generally nonzero for nonconvex problems. So far, most of the studies focus on continuous variables in fuzzy optimization problems. And, in real problems and models, fuzzy optimization problems also involve discrete and mixed variables. To address the above problems, we improve the extended duality theory by adding fuzzy objective functions. In this paper, we first define continuous fuzzy nonlinear programming problems, discrete fuzzy nonlinear programming problems, and mixed fuzzy nonlinear programming problems and then provide the extended dual problems, respectively. Finally we prove the weak and strong extended duality theorems, and the results show no duality gap between the original problem and extended dual problem.

scholarly journals A Novel Lagrangian Multiplier Update Algorithm for Short-Term Hydro-Thermal Coordination

Energies ◽  
2020 ◽  
Vol 13 (24) ◽  
pp. 6621
Author(s):  
P. M. R. Bento ◽  
S. J. P. S. Mariano ◽  
M. R. A. Calado ◽  
L. A. F. M. Ferreira
Keyword(s):  
Lagrange Multipliers ◽  
Large Scale ◽  
Dual Problem ◽  
Electrical Power ◽  
Decomposition Methods ◽  
Subgradient Methods ◽  
Lagrangian Multiplier ◽  
Step Size ◽  
Primal Problem ◽  
Require Time

The backbone of a conventional electrical power generation system relies on hydro-thermal coordination. Due to its intrinsic complex, large-scale and constrained nature, the feasibility of a direct approach is reduced. With this limitation in mind, decomposition methods, particularly Lagrangian relaxation, constitutes a consolidated choice to “simplify” the problem. Thus, translating a relaxed problem approach indirectly leads to solutions of the primal problem. In turn, the dual problem is solved iteratively, and Lagrange multipliers are updated between each iteration using subgradient methods. However, this class of methods presents a set of sensitive aspects that often require time-consuming tuning tasks or to rely on the dispatchers’ own expertise and experience. Hence, to tackle these shortcomings, a novel Lagrangian multiplier update adaptative algorithm is proposed, with the aim of automatically adjust the step-size used to update Lagrange multipliers, therefore avoiding the need to pre-select a set of parameters. A results comparison is made against two traditionally employed step-size update heuristics, using a real hydrothermal scenario derived from the Portuguese power system. The proposed adaptive algorithm managed to obtain improved performances in terms of the dual problem, thereby reducing the duality gap with the optimal primal problem.

Duality Theory

1995 ◽  
Author(s):  
Christodoulos A. Floudas
Keyword(s):  
Duality Theory ◽  
Dual Problem ◽  
Optimization Problems ◽  
Strong Duality ◽  
Perturbation Function ◽  
Primal Problem ◽  
Theory Section ◽  
Existence Conditions ◽  
Definition Of ◽  
Optimal Multiplier

Nonlinear optimization problems have two different representations, the primal problem and the dual problem. The relation between the primal and the dual problem is provided by an elegant duality theory. This chapter presents the basics of duality theory. Section 4.1 discusses the primal problem and the perturbation function. Section 4.2 presents the dual problem. Section 4.3 discusses the weak and strong duality theorems, while section 4.4 discusses the duality gap. This section presents the formulation of the primal problem, the definition and properties of the perturbation function, the definition of stable primal problem, and the existence conditions of optimal multiplier vectors.

A New Structural Optimization Method Based on Group Hunting of Animals: Hunting Search (HuS)

2010 ◽  
Author(s):  
R. Oftadeh ◽  
M. J. Mahjoob
Keyword(s):  
Structural Optimization ◽  
Numerical Optimization ◽  
Optimization Problems ◽  
Optimization Technique ◽  
Harmony Search ◽  
Optimization Methods ◽  
Optimization Method ◽  
Global Optimization Methods ◽  
Multi Level ◽  
Group Hunting

This paper presents a novel structural optimization algorithm based on group hunting of animals such as lions, wolves, and dolphins. Although these hunters have differences in the way of hunting but they are common in that all of them look for a prey in a group. The hunters encircle the prey and gradually tighten the ring of siege until they catch the prey. In addition, each member of the group corrects its position based on its own position and the position of other members. If the prey escapes from the ring, the hunters reorganize the group to siege the prey again. A benchmark numerical optimization problems is presented to show how the algorithm works. Three benchmark structural optimization problems are also presented to demonstrate the effectiveness and robustness of the proposed Hunting Search (HuS) algorithm for structural optimization. The objective in these problems is to minimize the weight of bar trusses. Both sizing and layout optimization variables are included, too. The proposed algorithm is compared with other global optimization methods such as CMLPSA (Corrected Multi-Level & Multi-Point Simulated Annealing) and HS (Harmony Search). The results indicate that the proposed method is a powerful search and optimization technique. It yields comparable and in some cases, better solutions compared to those obtained using current algorithms when applied to structural optimization problems.

scholarly journals A Hybrid Epigraph Directions Method for Nonsmooth and Nonconvex Constrained Optimization via Generalized Augmented Lagrangian Duality and a Genetic Algorithm

2018 ◽  
Vol 2018 ◽  
pp. 1-21
Author(s):  
Wilhelm P. Freire ◽  
Afonso C. C. Lemonge ◽  
Tales L. Fonseca ◽  
Hernando J. R. Franco
Keyword(s):  
Genetic Algorithm ◽  
Constrained Optimization ◽  
Augmented Lagrangian ◽  
Dual Problem ◽  
Optimization Problems ◽  
Lagrangian Duality ◽  
Dual Function ◽  
Constrained Optimization Problems ◽  
Nonconvex Constrained Optimization ◽  
Nonsmooth And Nonconvex Optimization

The Interior Epigraph Directions (IED) method for solving constrained nonsmooth and nonconvex optimization problem via Generalized Augmented Lagrangian Duality considers the dual problem induced by a Generalized Augmented Lagrangian Duality scheme and obtains the primal solution by generating a sequence of iterates in the interior of the epigraph of the dual function. In this approach, the value of the dual function at some point in the dual space is given by minimizing the Lagrangian. The first version of the IED method uses the Matlab routine fminsearch for this minimization. The second version uses NFDNA, a tailored algorithm for unconstrained, nonsmooth and nonconvex problems. However, the results obtained with fminsearch and NFDNA were not satisfactory. The current version of the IED method, presented in this work, employs a Genetic Algorithm, which is free of any strategy to handle the constraints, a difficult task when a metaheuristic, such as GA, is applied alone to solve constrained optimization problems. Two sets of constrained optimization problems from mathematics and mechanical engineering were solved and compared with literature. It is shown that the proposed hybrid algorithm is able to solve problems where fminsearch and NFDNA fail.

scholarly journals Economic Load Dispatch using Modified Vortex Search Algorithm

2019 ◽  
Vol 8 (2S11) ◽  
pp. 3395-3401
Keyword(s):  
Power Systems ◽  
Optimization Problems ◽  
Search Algorithm ◽  
Optimization Technique ◽  
Particle Swarm ◽  
Test System ◽  
Economic Load Dispatch ◽  
Step Size ◽  
Parameters Tuning ◽  
Generation Costs

This paper presents an efficacy optimization technique for solving Economic Load Dispatch (ELD) problem in power systems. Economic load dispatch has become more complicated with added ramp rate limits constraints to the problem. Latterly their were many traditional approach’s applied to ELD problem like Merit order loading, particle swarm optimation technique. This approach can be efficient for ELD problems. It cannot be applied due to high elaboration of their solutions. The Vortex Search Algorithm (VSA) is a newly advanced algorithm influenced by a vortex arrangement which can be designed as a number of nested circles. VSA approach was advanced from state of stirring liquids. The expediency and effectiveness of this approach is determined in different cases. It has no additional problem-specific parameters and it can be applicable to the optimization problems without control parameters tuning. VS Algorithm basically adjusts its step size automatically for the changing values of radius (circles) to improve the solution. In this proposed work, modified vortex search (MVSA) algorithm is applied to solve the ELD problem in some 6-unit test system by considering the system constraints and also the performance of this algorithm can be analyzed in terms of total generation costs and power losses. Obtained results of the test systems will be compared with particle swarm optimization (PSO), Merit order loading, VSA literature. The obtained results will demonstrate the MVSA algorithm is efficient way of solving the ELD problem and finding the output power of all the generation units accurately.

Pareto-Optimal Solutions for a Truss Problem

2011 ◽  
Vol 423 ◽  
pp. 53-64
Author(s):  
W. El Alem ◽  
A. El Hami ◽  
Rachid Ellaia
Keyword(s):  
Optimization Problems ◽  
Optimization Technique ◽  
Pareto Frontier ◽  
Optimization Method ◽  
Simultaneous Optimization ◽  
Multi Objective Optimization ◽  
Recent Contribution ◽  
Multi Objective ◽  
Structural Problems ◽  
Nbi Method

Most optimization problems, particularly those in engineering design, require the simultaneous optimization of more than one objective function. In this context, the solutions of these problems are based on the Pareto frontier construction. Substantial efforts have been made in recent years to develop methods for the construction of Pareto frontiers that guarantee uniform distribution and exclude the non-Pareto and local Pareto points. The Normal Boundary Intersection (NBI) is a recent contribution that generates a well-distributed Pareto frontier efficiently. Nevertheless, this method should be combined with a global optimization method to ensure the convergence to the global Pareto frontier. This paper proposes the NBI method using Adaptive Simulated Annealing (ASA) algorithm, namely NBI-ASA as a global nonlinear multi-objective optimization method. A well known benchmark multi-objective problem has been chosen from the literature to demonstrate the validity of the proposed method, applicability of the method for structural problems has been tested through a truss problem and promising results were obtained. The results indicate that the proposed method is a powerful search and multi-objective optimization technique that may yield better solutions to engineering problems than those obtained using current algorithms.

scholarly journals Improving optimization using adaptive algorithms

2021 ◽  
Vol 16 (1) ◽  
pp. 14-18
Author(s):  
László Kota ◽  
Károly Jármai
Keyword(s):  
Optimization Problems ◽  
Optimization Technique ◽  
Optimization Methods ◽  
Adaptive Methods ◽  
Optimization Method ◽  
Fine Tuning ◽  
Logistics Industry ◽  
Highly Nonlinear ◽  
Industrial Projects ◽  
Suboptimal Solution

AbstractIn the research projects and industrial projects severe optimization problems can be met, where the number of variables is high, there are a lot of constraints, and they are highly nonlinear and mostly discrete issues, where the running time can be calculated sometimes in weeks with the usual optimization methods on an average computer. In most cases in the logistics industry, the most robust constraint is the time. The optimizations are running on a typical office configuration, and the company accepts the suboptimal solution what the optimization method gives within the appropriate time limit. That is, why adaptivity is needed. The adaptivity of the optimization technique includes parameters of fine-tuning. On this way, the most sensitive setting can be found. In this article, some additional adaptive methods for logistic problems have been investigated to increase the effectivity, improve the solution in a strict time condition.

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