Convex Grey Optimization

Many optimization problems are formulated from a real scenario involving incomplete information due to uncertainty in reality. The uncertainties can be expressed with appropriate probability distributions or fuzzy numbers with a membership function, if enough information can be accessed for the construction of either the probability density function or the membership of the fuzzy numbers. However, in some cases there may not be enough information for that and grey numbers need to be used. A grey number is an interval number to represent the value of a quantity. Its exact value or the likelihood is not known but the maximum and/or the minimum possible values are. Applications in space exploration, robotics and engineering can be mentioned which involves such a scenario. An optimization problem is called a grey optimization problem if it involves a grey number in the objective function and/or constraint set. Unlike its wide applications, not much research is done in the field. Hence, in this paper, a convex grey optimization problem will be discussed. It will be shown that an optimal solution for a convex grey optimization problem is a grey number where the lower and upper limit are computed by solving the problem in an optimistic and pessimistic way. The optimistic way is when the decision maker counts the grey numbers as decision variables and optimize the objective function for all the decision variables whereas the pessimistic way is solving a minimax or maximin problem over the decision variables and over the grey numbers.

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An evolutionary-based optimization algorithm for truss sizing design

Vietnam Journal of Mechanics ◽

10.15625/0866-7136/7476 ◽

2016 ◽

Vol 38 (4) ◽

pp. 307-317

Author(s):

Pham Hoang Anh

Keyword(s):

Local Search ◽

Objective Function ◽

Optimization Algorithm ◽

Optimization Problems ◽

Optimal Solution ◽

The Other ◽

Truss Structures ◽

Benchmark Problems ◽

Discrete Variables ◽

Objective Functions

In this paper, the optimal sizing of truss structures is solved using a novel evolutionary-based optimization algorithm. The efficiency of the proposed method lies in the combination of global search and local search, in which the global move is applied for a set of random solutions whereas the local move is performed on the other solutions in the search population. Three truss sizing benchmark problems with discrete variables are used to examine the performance of the proposed algorithm. Objective functions of the optimization problems are minimum weights of the whole truss structures and constraints are stress in members and displacement at nodes. Here, the constraints and objective function are treated separately so that both function and constraint evaluations can be saved. The results show that the new algorithm can find optimal solution effectively and it is competitive with some recent metaheuristic algorithms in terms of number of structural analyses required.

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METHODS OF FORMULATING AND SOLVING OPTIMIZATION PROBLEMS OF CUTTING LOGS OF LARGE SIZE BY BAR-SAWING METHOD WITH CUTTING OUT ONE BAR AND FIVE PAIRS OF SIDE EDGING BOARDS WITH SUBSEQUENT SAWING LUMBER FOR EDGED BOARDS

Forestry Engineering Journal ◽

10.12737/25204 ◽

2017 ◽

Vol 7 (1) ◽

pp. 137-150

Author(s):

Агапов ◽

Aleksandr Agapov

Keyword(s):

Mathematical Model ◽

Objective Function ◽

Optimization Problem ◽

Optimization Problems ◽

Target Function ◽

Cross Sectional ◽

Optimization Task ◽

Cutting Out ◽

The Mathematical Model ◽

The Relationship

For the first time the mathematical model of task optimization for this scheme of cutting logs, including the objective function and six equations of connection. The article discusses Pythagorean area of the logs. Therefore, the target function is represented as the sum of the cross-sectional areas of edging boards. Equation of the relationship represents the relationship of the diameter of the logs in the vertex end with the size of the resulting edging boards. This relationship is described through the use of the Pythagorean Theorem. Such a representation of the mathematical model of optimization task is considered a classic one. However, the solution of this mathematical model by the classic method is proved to be problematic. For the solution of the mathematical model we used the method of Lagrange multipliers. Solution algorithm to determine the optimal dimensions of the beams and side edging boards taking into account the width of cut is suggested. Using a numerical method, optimal dimensions of the beams and planks are determined, in which the objective function takes the maximum value. It turned out that with the increase of the width of the cut, thickness of the beam increases and the dimensions of the side edging boards reduce. Dimensions of the extreme side planks to increase the width of cut is reduced to a greater extent than the side boards, which are located closer to the center of the log. The algorithm for solving the optimization problem is recommended to use for calculation and preparation of sawing schedule in the design and operation of sawmill lines for timber production. When using the proposed algorithm for solving the optimization problem the output of lumber can be increased to 3-5 %.

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Robust Approximate Optimality Conditions for Uncertain Nonsmooth Optimization with Infinite Number of Constraints

Mathematics ◽

10.3390/math7010012 ◽

2018 ◽

Vol 7 (1) ◽

pp. 12 ◽

Cited By ~ 7

Author(s):

Xiangkai Sun ◽

Hongyong Fu ◽

Jing Zeng

Keyword(s):

Optimality Conditions ◽

Optimization Problem ◽

Constraint Qualification ◽

Optimization Problems ◽

Optimization Technique ◽

Optimal Solution ◽

Sufficient Optimality Conditions ◽

Convex Constraint ◽

Approximate Optimality Conditions ◽

Necessary And Sufficient

This paper deals with robust quasi approximate optimal solutions for a nonsmooth semi-infinite optimization problems with uncertainty data. By virtue of the epigraphs of the conjugates of the constraint functions, we first introduce a robust type closed convex constraint qualification. Then, by using the robust type closed convex constraint qualification and robust optimization technique, we obtain some necessary and sufficient optimality conditions for robust quasi approximate optimal solution and exact optimal solution of this nonsmooth uncertain semi-infinite optimization problem. Moreover, the obtained results in this paper are applied to a nonsmooth uncertain optimization problem with cone constraints.

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An Interactive Neural Network for Constrained Multi-Objective Optimization with Application to the Design of Digital Filters

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.433-440.2808 ◽

2012 ◽

Vol 433-440 ◽

pp. 2808-2816

Author(s):

Jian Jin Zheng ◽

You Shen Xia

Keyword(s):

Neural Network ◽

Neural Networks ◽

Computational Complexity ◽

Digital Filters ◽

Optimization Problem ◽

Optimization Problems ◽

Optimal Solution ◽

Multi Objective Optimization ◽

Multi Objective ◽

Single Objective

This paper presents a new interactive neural network for solving constrained multi-objective optimization problems. The constrained multi-objective optimization problem is reformulated into two constrained single objective optimization problems and two neural networks are designed to obtain the optimal weight and the optimal solution of the two optimization problems respectively. The proposed algorithm has a low computational complexity and is easy to be implemented. Moreover, the proposed algorithm is well applied to the design of digital filters. Computed results illustrate the good performance of the proposed algorithm.

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Linear Programming and Fuzzy Optimization to Substantiate Investment Decisions in Tangible Assets

Entropy ◽

10.3390/e22010121 ◽

2020 ◽

Vol 22 (1) ◽

pp. 121

Author(s):

Marcel-Ioan Boloș ◽

Ioana-Alexandra Bradea ◽

Camelia Delcea

Keyword(s):

Linear Programming ◽

Objective Function ◽

Hybrid Model ◽

Fuzzy Numbers ◽

Investment Decision ◽

Fuzzy Optimization ◽

Simplex Algorithm ◽

Fuzzy Variables ◽

Decision Variables ◽

Tangible Assets

This paper studies the problem of tangible assets acquisition within the company by proposing a new hybrid model that uses linear programming and fuzzy numbers. Regarding linear programming, two methods were implemented in the model, namely: the graphical method and the primal simplex algorithm. This hybrid model is proposed for solving investment decision problems, based on decision variables, objective function coefficients, and a matrix of constraints, all of them presented in the form of triangular fuzzy numbers. Solving the primal simplex algorithm using fuzzy numbers and coefficients, allowed the results of the linear programming problem to also be in the form of fuzzy variables. The fuzzy variables compared to the crisp variables allow the determination of optimal intervals for which the objective function has values depending on the fuzzy variables. The major advantage of this model is that the results are presented as value ranges that intervene in the decision-making process. Thus, the company’s decision makers can select any of the result values as they satisfy two basic requirements namely: minimizing/maximizing the objective function and satisfying the basic requirements regarding the constraints resulting from the company’s activity. The paper is accompanied by a practical example.

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SURFACE EFFECT CORRECTION OF MOISTURE DETERMINATION BY NEUTRON PROBE USING PSO TECHNIQUE AND MCNP

International Journal of Computational Methods ◽

10.1142/s0219876209001802 ◽

2009 ◽

Vol 06 (02) ◽

pp. 247-255

Author(s):

ALI ASGHAR MOWLAWI ◽

HADI SADOGHI YAZDI ◽

MEHDI ARGHIANI ◽

JABER ROOHI ◽

RAHIM KOOHI-FAYEGH ◽

...

Keyword(s):

Optimization Problem ◽

Optimization Problems ◽

Surface Effect ◽

Soil Surface ◽

Moisture Determination ◽

Swarm Optimization ◽

Convex Optimization Problem ◽

Neutron Probe ◽

Decision Variables ◽

Paraffin Block

The usefulness of the neutron meter is limited for moisture measurement near the soil surface. In this present work, optimum dimension of a paraffin block has been calculated to correct the surface effect in order to use neutron probe near the soil surface by MCNP4C code and Particle Swarm Optimization (PSO) technique. PSO is chiefly a technique to find a global or quasi-minimum for a nonlinear and non-convex optimization problem, and there have been few studies into optimization problems with discrete decision variables. The results show a paraffin block 23.55 × 23.55 cm2 square base with 4.84, 4.92, 5.10, 5.23, and 5.47 cm thickness which can correct the surface effect fairly for 0.10, 0.20, 0.30, 0.40, and 0.50 g/g moisture.

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Novel Interior Point Algorithms for Solving Nonlinear Convex Optimization Problems

Advances in Operations Research ◽

10.1155/2015/487271 ◽

2015 ◽

Vol 2015 ◽

pp. 1-7

Author(s):

Sakineh Tahmasebzadeh ◽

Hamidreza Navidi ◽

Alaeddin Malek

Keyword(s):

Convex Optimization ◽

Objective Function ◽

Interior Point ◽

Optimization Problems ◽

Numerical Algorithms ◽

Optimal Solution ◽

Linear Constraints ◽

Convex Optimization Problems ◽

Novel Algorithms ◽

Interior Point Technique

This paper proposes three numerical algorithms based on Karmarkar’s interior point technique for solving nonlinear convex programming problems subject to linear constraints. The first algorithm uses the Karmarkar idea and linearization of the objective function. The second and third algorithms are modification of the first algorithm using the Schrijver and Malek-Naseri approaches, respectively. These three novel schemes are tested against the algorithm of Kebiche-Keraghel-Yassine (KKY). It is shown that these three novel algorithms are more efficient and converge to the correct optimal solution, while the KKY algorithm fails in some cases. Numerical results are given to illustrate the performance of the proposed algorithms.

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Model Building and Optimization Analysis of MDF Continuous Hot-Pressing Process by Neural Network

Mathematical Problems in Engineering ◽

10.1155/2016/1327235 ◽

2016 ◽

Vol 2016 ◽

pp. 1-16

Author(s):

Qingfa Li ◽

Yaqiu Liu ◽

Liangkuan Zhu

Keyword(s):

Neural Network ◽

Objective Function ◽

Convex Subset ◽

Hot Pressing ◽

Optimization Problem ◽

Model Building ◽

Optimization Problems ◽

Initial Point ◽

Data Set ◽

Pressing Process

We propose a one-layer neural network for solving a class of constrained optimization problems, which is brought forward from the MDF continuous hot-pressing process. The objective function of the optimization problem is the sum of a nonsmooth convex function and a smooth nonconvex pseudoconvex function, and the feasible set consists of two parts, one is a closed convex subset ofRn, and the other is defined by a class of smooth convex functions. By the theories of smoothing techniques, projection, penalty function, and regularization term, the proposed network is modeled by a differential equation, which can be implemented easily. Without any other condition, we prove the global existence of the solutions of the proposed neural network with any initial point in the closed convex subset. We show that any accumulation point of the solutions of the proposed neural network is not only a feasible point, but also an optimal solution of the considered optimization problem though the objective function is not convex. Numerical experiments on the MDF hot-pressing process including the model building and parameter optimization are tested based on the real data set, which indicate the good performance of the proposed neural network in applications.

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A Progressive Approximation Approach for the Exact Solution of Sparse Large-Scale Binary Interdiction Games

INFORMS Journal on Computing ◽

10.1287/ijoc.2021.1085 ◽

2021 ◽

Author(s):

Claudio Contardo ◽

Jorge A. Sefair

Keyword(s):

Exact Solution ◽

Approximation Algorithm ◽

Facility Location ◽

Shortest Path ◽

Large Scale ◽

Optimization Problem ◽

Optimal Solution ◽

Solution Space ◽

Decision Variables ◽

Algorithmic Framework

We present a progressive approximation algorithm for the exact solution of several classes of interdiction games in which two noncooperative players (namely an attacker and a follower) interact sequentially. The follower must solve an optimization problem that has been previously perturbed by means of a series of attacking actions led by the attacker. These attacking actions aim at augmenting the cost of the decision variables of the follower’s optimization problem. The objective, from the attacker’s viewpoint, is that of choosing an attacking strategy that reduces as much as possible the quality of the optimal solution attainable by the follower. The progressive approximation mechanism consists of the iterative solution of an interdiction problem in which the attacker actions are restricted to a subset of the whole solution space and a pricing subproblem invoked with the objective of proving the optimality of the attacking strategy. This scheme is especially useful when the optimal solutions to the follower’s subproblem intersect with the decision space of the attacker only in a small number of decision variables. In such cases, the progressive approximation method can solve interdiction games otherwise intractable for classical methods. We illustrate the efficiency of our approach on the shortest path, 0-1 knapsack and facility location interdiction games. Summary of Contribution: In this article, we present a progressive approximation algorithm for the exact solution of several classes of interdiction games in which two noncooperative players (namely an attacker and a follower) interact sequentially. We exploit the discrete nature of this interdiction game to design an effective algorithmic framework that improves the performance of general-purpose solvers. Our algorithm combines elements from mathematical programming and computer science, including a metaheuristic algorithm, a binary search procedure, a cutting-planes algorithm, and supervalid inequalities. Although we illustrate our results on three specific problems (shortest path, 0-1 knapsack, and facility location), our algorithmic framework can be extended to a broader class of interdiction problems.

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Biobjective Optimization of Well Placement: Algorithm, Validation, and Field Testing

SPE Journal ◽

10.2118/203960-pa ◽

2021 ◽

pp. 1-28

Author(s):

Faruk Alpak ◽

Vivek Jain ◽

Yixuan Wang ◽

Guohua Gao

Keyword(s):

Multiobjective Optimization ◽

Objective Function ◽

Pareto Front ◽

Optimization Problem ◽

Optimization Problems ◽

Field Testing ◽

Efficient Implementation ◽

Field Development ◽

Biobjective Optimization ◽

Single Objective

Summary We describe the development and validation of a novel algorithm for field-development optimization problems and document field-testing results. Our algorithm is founded on recent developments in bound-constrained multiobjective optimization of nonsmooth functions for problems in which the structure of the objective functions either cannot be exploited or are nonexistent. Such situations typically arise when the functions are computed as the result of numerical modeling, such as reservoir-flow simulation within the context of field-development planning and reservoir management. We propose an efficient implementation of a novel parallel algorithm, namely BiMADS++, for the biobjective optimization problem. Biobjective optimization is a special case of multiobjective optimization with the property that Pareto points may be ordered, which is extensively exploited by the BiMADS++ algorithm. The optimization algorithm generates an approximation of the Pareto front by solving a series of single-objective formulations of the biobjective optimization problem. These single-objective problems are solved using a new and more efficient implementation of the mesh adaptive direct search (MADS) algorithm, developed for nonsmooth optimization problems that arise within reservoir-simulation-based optimization workflows. The MADS algorithm is extensively benchmarked against alternative single-objective optimization techniques before the BiMADS++ implementation. Both the MADS optimization engine and the master BiMADS++ algorithm are implemented from the ground up by resorting to a distributed parallel computing paradigm using message passing interface (MPI) for efficiency in industrial-scaleproblems. BiMADS++ is validated and field tested on well-location optimization (WLO) problems. We first validate and benchmark the accuracy and computational performance of the MADS implementation against a number of alternative parallel optimizers [e.g., particle-swarm optimization (PSO), genetic algorithm (GA), and simultaneous perturbation and multivariate interpolation (SPMI)] within the context of single-objective optimization. We also validate the BiMADS++ implementation using a challenging analytical problem that gives rise to a discontinuous Pareto front. We then present BiMADS++ WLO applications on two simple, intuitive, and yet realistic problems, and a model for a real problem with known Pareto front. Finally, we discuss the results of the field-testing work on three real-field deepwater models. The BiMADS++ implementation enables the user to identify various compromise solutions of the WLO problem with a single optimization run without resorting to ad hoc adjustments of penalty weights in the objective function. Elimination of this “trial-and-error” procedure and distributed parallel implementation renders BiMADS++ easy to use and significantly more efficient in terms of computational speed needed to determine alternative compromise solutions of a given WLO problem at hand. In a field-testing example, BiMADS++ delivered a workflow speedup of greater than fourfold with a single biobjective optimization run over the weighted-sumsobjective-function approach, which requires multiple single-objective-function optimization runs.

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