An optimization method to solve a fully intuitionistic fuzzy non-linear separable programming problem

This paper develops an approach for solving non-linear programming problem in intuitionistic fuzzy environment. Since, fuzziness may occur in some or all parameters of a system, we discuss both situations in this study. We transform the intuitionisti

The concept of duality plays an important role in optimization theory. This paper discusses some relations between primal and dual nonlinear programming problems in fuzzy environment. Here, fuzzy feasible region for a general fuzzy nonlinear programming is formed and the concept of fuzzy feasible solution is defined. First order dual relation for fuzzy nonlinear programming problem is studied.

Converting general nonlinear programming problems into separable programming problems with feedforward neural networks

A genetic algorithm (GA) based approach to solve a fuzzy multi-objective nonlinear programming (FMNP) problem is presented and applied to water resources allocation. A FMNP model of regional water allocation is established originally. The objective functions of the model include three conflicting goals which are economic profits, environment goal and society goal. By turning the FMNP to an optimal level cut problem, the ldquofuzzy optimal solutionrdquo is solved by using a genetic algorithm. An example of the Dongyingpsilas water allocation strategy is provided by using the proposed method.

An integrated computer programme in Fortran IV for continuous or discrete non linear programming problems is presented. Several recent techniques and algorithms for non-linear programming have been adapted and new ideas have been introduced. They include the minimax and exterior-point approaches to non-linear programming, least pth optimization and the Dakin tree-search algorithm. The user may optionally choose the combination of techniques and algorithms best suited to his problems. Since many practical design problems can be easily formulated as non-linear programming problems, the programme, called DISOPT, enjoys a very wide range of applications such as continuous and discrete tolerance assignments, digital filter design, circuit design, system modelling and approximation problems. Numerical results for a number of functions and circuit tolerance optimization problems are presented in this paper to demonstrate the performance of DISOPT.

A nonlinear programming approach to kinematic shakedown analysis of frictional materials

Constrained optimization for microbial fermentation was studied. For optimization, we used not the maximum principle but a nonlinear programming method because of the need to consider many metabolic reactions. In the case of L-lysine fermentation, the optimization problem in L-lysine production was formulated as a nonlinear programming problem. In general, the state equations based on material balances are represented as differential equations, but such equations which are dependent on time can not be applied to a nonlinear programming problem. Therefore, the state equations were made discrete in a time base, and a new single vector which is not dependent on time was substituted. From these formulae, the objective function and the constraints using nonlinear programming problem were defined as the amount of L-lysine produced, and as a metabolic reaction model and empirical equations, respectively. Computer program was developed to solve this constrained nonlinear programming problem. The applied algorithm of the computer programming was a sequential quadratic programming method (SQP method). When the constrained nonlinear programming problem is solved using the SQP method, the maximum amount of L-lysine produced and the optimal feeding rate of L-threonine could be calculated. From the calculated results, it was clear that introduction of the equality and inequality constraints was easy. L-Lysine at a concentration up to 75.3 g/l could be produced when the fermentation was carried out under optimal conditions.

The objective of this paper is to develop matrix games with pay-offs of triangular hesitant fuzzy elements (THFEs). To solve such games, a new methodology has been derived based on the notion of weighted average operator and score function of THFEs. Firstly, we formulate two non-linear programming problems with THFEs. Then applying the score function of THFEs, we transform these two problems into two non-linear multi-objective programming problems with triangular fuzzy numbers (TFNs). Finally, the Lexicographic method is used to solve these two multi-objective programming problems. A market share problem is considered to show the validity and applicability of the proposed methodology.

We consider the problem of joint optimal placement of phasor measurement units (PMU) and conventional measurements to ensure full observability in power systems. The formulation is initially posed as a nonlinear integer programming problem and then transformed in to an equivalent integer linear programming (ILP) problem by introducing auxiliary variables and constraints. The resulting ILP problem is solved for the optimal solution on IEEE 14-, 57-, and 118-bus systems considering zero-injection buses. To extend the formulation to large-scale systems, two heuristics are proposed where the nonlinear problem is decomposed into two separate problems of lesser complexity. The heuristics are evaluated on standard IEEE test cases and a large 2383-bus Polish system. The heuristics yield solutions close to optimal (difference of 1 PMU) solutions on standard IEEE systems, while substantially reducing problem complexity and run time. The placement results obtained with the proposed formulation require fewer PMUs for observability compared to systems with fixed locations of conventional measurements. The results thus potentially provide a more economical solution to system observability compared to those obtained solely with PMU placement.

: Several algorithms for solving problems in linear, quadratic, and nonlinear programming, network flows, and facilities location were developed and analyzed. Results include: (1) The analysis of the computational complexity of the problem of determining an optimally sparse representation of the null space of a matrix, and the development of worst-case bounds for the shadow-vertex simplex algorithm and several heuristics for distance constrained discrete facility location problems and conditional covering problems; (2) The development of efficient algorithms for dense and sparse assignment problems, strictly convex quadratic programming problems, and a nonlinear programming problem that arises when maximizing a correlation coefficient subject to linear constraints; and (3) The application of iterative methods to large sparse equality-constraind quadratic programs and the development of multiple constraint deletion strategies for active-set algorithms for linearly consrained nonlinear programming problems. (Author)

Objective: The purpose of the work is to analyze, study the processes of digital transformation and make optimal decisions in economic and engineering systems based on the theories and methods of vector optimization. Methods: Methods for studying the processes of digital transformation of economic and engineering systems are based on the author's theory of vector optimization, which is represented by: axiomatics and optimality principles that show how one solution is better than another. On the basis of the principles of optimality, constructive methods for solving vector problems have been built, firstly, with equivalent criteria, and, secondly, with a given priority of the criterion: https://rdcu.be/bhZ8i (Springer Nature's "Vector optimization with equivalent and priority criteria" is distributed free of charge.) The paper presents that the problem of accepting the optimal process is carried out as a result of the digital transformation of the initial (statistical, experimental) data into vector problems in optimization, which represents a mathematical model of economic and engineering systems. Depending on the number and type of criteria, we solve problems with a vector of criteria, each component of which is directed to the maximum, and with a vector of criteria, each component of which is aimed at a minimum, in aggregate. Results: The result of the study is the practical orientation of digital transformation, which is shown in the computer-aided design of economic and engineering systems based on vector optimization. For this purpose, software for solving vector problems of linear and nonlinear programming has been developed. Linear vector problems are used in economic systems (forecasting, planning). Software for solving vector problems of nonlinear programming are used digital transformation of optimal decision-making in engineering problems. Conclusion: In conclusion, numerical examples in the field of economic systems are presented by the digital transformation of optimal decision-making in the development of the company. When making optimal decisions in engineering systems, the following has been developed: the construction of initial data (technical task) for modeling; transformation of a mathematical model under uncertainty into a model under conditions of certainty; making an optimal decision with equivalent criteria; making the optimal decision with a given priority criterion

The paper claims that the primary importance in solving the classification problem is to find the conditions for dividing the General complexity into classes, determine the quality of such a bundle, and verify the classifier model. We consider a mathematical model of a non-randomized classifier of features obtained without a teacher, when the number of classes is not set a priori, but only its upper bound is set. The mathematical model is presented in the form of a statement of a minimax conditional extreme task, and it is a problem of searching for the matrix of belonging of objects to a class, and representative (reference) elements within each class. The development of the feature classifier is based on the synthesis of two-dimensional probability density in the coordinate space: classes-objects. Using generalized functions, the probabilistic problem of finding the minimum Bayesian risk is reduced to a deterministic problem on a set of non-randomized classifiers. At the same time, the use of specially introduced constraints fixes non-randomized decision rules and plunges the integer problem of nonlinear programming into a General continuous nonlinear problem. For correct synthesis of the classifier, the dispersion curve of the isotropic sample is necessary. It is necessary to use the total intra-class and inter-class variance to characterize the quality of classification. The classification problem can be interpreted as a particular problem of the theory of catastrophes. Under the conditions of limited initial data, a minimax functional was found that reflects the quality of classification for a quadratic loss function. The developed mathematical model is classified as an integer nonlinear programming problem. The model is given using polynomial constraints to the form of a General problem of nonlinear continuous programming. The necessary conditions for the bundle into classes are found. These conditions can be used as sufficient when testing the hypothesis about the existence of classes.

On an exact penalty function method for nonlinear mixed discrete programming problems and its applications in search engine advertising problems

Nonlinear optimization is becoming challengeable in information sciences and various industrial applications, but nonlinear problems solved by the classical particle swarm-based methods are usually characterized with low efficiency, accuracy, and convergence speed in specific issues. To solve these problems and enhance the nonlinear optimization performance, an improved metaheuristic particle swarm optimization (PSO) model is proposed here. First, the optimization principles and model of the new method are introduced, and algorithms of the improved PSO are presented. Then, influence of the model parameters, input dimensions, and different nonlinear problems on the PSO optimization characterizations are studied by Pareto set solving and optimization performance comparison. The analysis regarding diverse nonlinear problems and optimization methods demonstrates the feasibility of the proposed method. Finally, the performance evaluation is exhibited by the case study of nonlinear parameter optimization, CEC benchmark problems, and rank sum test, which all verify its effectiveness and reliability, as well as the significance and great application promise. Additionally, the future work is discussed.

This article considers bilevel linear programming problems where the coefficients of the objective functions and the constraints in the problem are given as fuzzy parameters. Stackelberg problems and Stackelberg solutions under fuzziness have not been fully discussed so far. New Stackelberg solution concepts under fuzziness, called possibilistic Stackelberg solutions , are proposed by incorporating possibility theory into Stackelberg solutions in non-fuzzy environments. This paper shows some theoretical properties and theorems on the formulated fuzzy bilevel programming problems. It is shown that the proposed possibilistic Stackelberg solutions are obtained by using linear or nonlinear programming techniques. The effectiveness of the proposed models is demonstrated by numerical experiments using transportation problems where the upper-level decision maker is a buyer and the lower-level one is a supplier.