scholarly journals On an extremum problem

1987 ◽  
Vol 28 (3) ◽  
pp. 376-392 ◽  
Author(s):  
Joanna Matula
Keyword(s):  
Existence Theorem ◽  
Optimization Problem ◽  
Integral Functional ◽  
Integral Form ◽  
Extremum Problem ◽  
Necessary Condition

AbstractWe consider an optimization problem in which the function being minimized is the sum of the integral functional and the full variation of control. For this problem, we prove the existence theorem, a necessary condition in an integral form and a local necessary condition in the case of monotonic controls.

scholarly journals A Kind of New Higher-Order Mond-Weir Type Duality for Set-Valued Optimization Problems

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 372
Author(s):  
Liu He ◽  
Qi-Lin Wang ◽  
Ching-Feng Wen ◽  
Xiao-Yan Zhang ◽  
Xiao-Bing Li
Keyword(s):  
Existence Theorem ◽  
Lower Order ◽  
Optimization Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Strong Duality ◽  
Higher Order ◽  
Weak Duality ◽  
Duality Theorems ◽  
Benson Proper Efficiency

In this paper, we introduce the notion of higher-order weak adjacent epiderivative for a set-valued map without lower-order approximating directions and obtain existence theorem and some properties of the epiderivative. Then by virtue of the epiderivative and Benson proper efficiency, we establish the higher-order Mond-Weir type dual problem for a set-valued optimization problem and obtain the corresponding weak duality, strong duality and converse duality theorems, respectively.

scholarly journals A non-existence theorem for (v, k,λ)-graphs

1970 ◽  
Vol 11 (3) ◽  
pp. 381-383 ◽  
Author(s):  
W. D. Wallis
Keyword(s):  
Existence Theorem ◽  
Necessary Condition ◽  
Image Position ◽  
Set Of Points

A (ν, κ, λ)-graph is defined in [3] as a graph on ν points, each of valency κ, and such that for any two points P and Q there are exactly λ points which are joined to both. In other words, if Si is the set of points joined to Pi, thenSi has k elements for any iSiSj has λ elements if i≠jThe sets Si are the blocks of a (v, k, λ)-configuration, so a necessary condition on v, k, and λ that a graph should exist is that a (v, k, λ)- configuration should exist. Another necessary condition, reported by Bose (see [1]) and others, is that there should be an integer m satisfying have equal parity. We shall prove that these conditions are not sufficient.

scholarly journals INFORMATION TECHNOLOGY OF TERRITORY COVERING BY SENSORS WITH THE CONSTANT INTERSECTION LEVEL AND COST MINIMIZATION

2020 ◽  
pp. 65-72
Author(s):  
V.Y. Petrivskyi ◽  
V.L. Shevchenko ◽  
O.S. Bychkov ◽  
V.M. Loza
Keyword(s):  
Information Technology ◽  
Optimization Problem ◽  
Data Transfer ◽  
Cost Minimization ◽  
Rapid Development ◽  
Computer Experiments ◽  
Necessary Condition ◽  
Energy Costs ◽  
Sensor Coverage ◽  
Activity Sensors

Thanks to the rapid development of technologies, in particular information, sensors have become widespread and used in all areas of human activity. Sensors and sensor networks have received special use during the collection and processing of data of various types. When monitoring a certain territory, the problem arises of its maximum coverage in order to increase the information content and completeness of the accumulated data. Simultaneously with the predominance of autonomous use of sensors, the problem of the duration of the sensor operation arises. This value depends on the capacity of the battery. In turn, engineers are faced with the task of minimizing the design of the sensors, which results in a decrease in the volume of the battery simultaneously with all other components. It is also obvious that as the sensor coverage radius increases, the energy consumption increases, which in turn shortens the sensor life. In addition to energy costs, the article considers the costs of servicing and purchasing sensors. Thus, in addition to maximizing the percentage of coverage of the study area, the problem of minimizing the total costs arises. Obviously, to ensure data transfer between sensors, a necessary condition is the presence of the intersection of the sensor coverage areas. In this case, the constant value of this parameter is considered. The materials propose an approach to solving the problem of maximizing the coverage of the territory with minimizing costs for a given level of intersection of the coverage areas of the sensors. The proposed approach is based on solving a nonlinear multiobjective optimization problem. Also, one of the options for solving the described problem is proposed to reduce the objective functions in one by using a weighted convolution of criteria. In addition, the article proposes an iterative approach to solving the described problem. A number of computer experiments have been carried out. The results of the performed computational experiments confirm the possibility of using the proposed information technology both in the form of an optimization problem and in the form of an iterative process.

Monotonicity Analysis of Taguchi’s Robust Circuit Design Problem

1990 ◽  
Author(s):  
Douglass J. Wilde
Keyword(s):  
Circuit Design ◽  
Design Problem ◽  
Recent Article ◽  
Optimization Problem ◽  
Algebraic Function ◽  
Necessary Condition ◽  
Control Range ◽  
Design Variables ◽  
Numerical Iteration ◽  
Monotonicity Analysis

Abstract A recent article showed how to formulate Taguchi’s robust circuit design problem rigorously as an optimization problem. A necessary condition for optimality was found to be that the control range be centered about the target value. This generates a constraint on the two design variables which cannot be solved for either variable. The present article shows that by approximating this unsolvable constraint with a simpler constraint that is solvable, one variable can be eliminated and the problem reduced to an unconstrained one in a single variable. Since this reduced objective turns out to be monotonic in the remaining design variable, its optimum value must be at the limit of its range. The corresponding optimum value of the other variable is then determined exactly from the true, not approximate, constraint. Since no model construction, experimentation, statistical analysis or numerical iteration is needed, this procedure is recommended whenever the input-output relation is known to be a monotonic algebraic function.

scholarly journals On G-invexity-type nonlinear programming problems

2015 ◽  
Vol 5 (1) ◽  
pp. 13-20
Author(s):  
Tadeusz Antczak ◽  
Manuel Arana Jiménez
Keyword(s):  
Optimization Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Inequality Constraints ◽  
Extremum Problem ◽  
Sufficient Optimality Conditions ◽  
New Class ◽  
Necessary And Sufficient ◽  
Wolfe Dual

In this paper, we introduce the concepts of KT-G-invexity and WD$-G-invexity for the considered differentiable optimization problem with inequality constraints. Using KT-G-invexity notion, we prove new necessary and sufficient optimality conditions for a new class of such nonconvex differentiable optimization problems. Further, the so-called G-Wolfe dual problem is defined for the considered extremum problem with inequality constraints. Under WD-G-invexity assumption, the necessary and sufficient conditions for weak duality between the primal optimization problem and its G-Wolfe dual problem are also established.

scholarly journals A new approach to optimality in a class of nonconvex smooth optimization problems

2018 ◽  
Vol 34 (1) ◽  
pp. 01-07
Author(s):  
TADEUSZ ANTCZAK ◽  
Keyword(s):  
Approximation Method ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Extremum Problem ◽  
Optimal Solutions ◽  
New Approach ◽  
Smooth Optimization

In this paper, a new approximation method for a characterization of optimal solutions in a class of nonconvex differentiable optimization problems is introduced. In this method, an auxiliary optimization problem is constructed for the considered nonconvex extremum problem. The equivalence between optimal solutions in the considered differentiable extremum problem and its approximated optimization problem is established under (Φ, ρ)-invexity hypotheses.

scholarly journals Optimization of Bridge Crane Movement Control

2018 ◽  
Vol 17 (5) ◽  
pp. 413-420
Author(s):  
V. S. Loveikin ◽  
Y. A. Romasevich
Keyword(s):  
Electric Drive ◽  
Optimization Problem ◽  
Dynamic Properties ◽  
Analytical Form ◽  
Energy Performance ◽  
Suboptimal Control ◽  
Electrical Performance ◽  
Necessary Condition ◽  
Bridge Crane ◽  
Phase Coordinates

Transient modes of bridge cranes movement determine their energy, dynamic and electrical performance, as well as productivity and durability of work. An optimal control problem of its movement has been solved while making an analysis of indicators for efficient performance of a bridge crane. Terminal and integral criteria have been selected as optimization criteria. They represent undesirable dynamic properties of the crane. Legendre method has been used to determine the possibility for achieving minimum of the optimization criterion. An analysis of the Euler-Poisson equation, which is a necessary condition for the minimum of the integral criterion, has shown that it is impossible to find a solution for the optimization problem in an analytical form. A method of differential evolution has been used in order to find an approximate solution to the optimization problem. The approximate (suboptimal) solution has been found in the complex domain, which is a limited domain conjunction of dynamic parameters and phase coordinates of the system. Limitation in the domain of the system phase coordinates (a polynomial basis function has been used in the paper) provides the possibility to attain absolute minimums of terminal problem criteria. A simulation of the bridge crane motion has been carried out in order to establish an efficiency for implementation of the suboptimal control. During this process dynamic mechanical characteristics of its electric drive have been taken into account. While carrying out the simulation, a frequency and an amplitude of the electric drive voltage in the crane movement mechanism have been changed (frequency scalar method for speed changing of an asynchronous electric drive has been used). A comparative analysis of the dynamic, kinematic, electrical and energy performance indicators of the bridge crane under suboptimal and S-curved (standard) laws of frequency and voltage variations in the crane electric drive has made it possible to establish an improvement in the efficiency of its operation under suboptimal control.

Some sharp results about the global existence and blowup of solutions to a class of pseudo-parabolic equations

2017 ◽  
Vol 147 (6) ◽  
pp. 1311-1331 ◽  
Author(s):  
Xiaoli Zhu ◽  
Fuyi Li ◽  
Yuhua Li
Keyword(s):  
Weak Solution ◽  
Global Existence ◽  
Parabolic Equations ◽  
Integral Form ◽  
Energy Functional ◽  
Nehari Manifold ◽  
Necessary Condition ◽  
Blowup Of Solutions ◽  
Decay Behaviour ◽  
Whole Space

In this paper we are interested in a sharp result about the global existence and blowup of solutions to a class of pseudo-parabolic equations. First, we represent a unique local weak solution in a new integral form that does not depend on any semigroup. Second, with the help of the Nehari manifold related to the stationary equation, we separate the whole space into two components S+ and S– via a new method, then a sufficient and necessary condition under which the weak solution blows up is established, that is, a weak solution blows up at a finite time if and only if the initial data belongs to S–. Furthermore, we study the decay behaviour of both the solution and the energy functional, and the decay ratios are given specifically.

Monotonicity Analysis of Taguchi’s Robust Circuit Design Problem

1992 ◽  
Vol 114 (4) ◽  
pp. 616-619 ◽  
Author(s):  
D. J. Wilde
Keyword(s):  
Circuit Design ◽  
Design Problem ◽  
Optimization Problem ◽  
Algebraic Function ◽  
The Other ◽  
Necessary Condition ◽  
Control Range ◽  
Design Variables ◽  
Numerical Iteration ◽  
Monotonicity Analysis

Taguchi’s robust circuit design problem can be formulated rigorously as an optimization problem. A necessary condition for optimality is that the control range be centered about the target value. This generates a constraint on the two design variables which cannot be solved for either variable. The present article shows that by approximating this unsolvable constraint with a simpler constraint that is solvable, one variable can be eliminated and the problem reduced to an unconstrained one in a single variable. Since this reduced objective turns out to be monotonic in the remaining design variable, its optimum value must be at the limit of its range. The corresponding optimum value of the other variable is then determined exactly from the true, not approximate, constraint. Since no model construction, experimentation, statistical analysis, or numerical iteration is needed, this procedure is recommended whenever the input-output relation is known to be a monotonic algebraic function.

scholarly journals Data-Driven Self-Optimizing Control: Unconstrained Optimization Problem

2021 ◽  
Vol 6 (3) ◽  
Author(s):  
Alhaji S Grema ◽  
Yi Cao ◽  
Modu B Grema
Keyword(s):  
Finite Difference ◽  
Unconstrained Optimization ◽  
Optimization Problem ◽  
Difference Schemes ◽  
Necessary Condition ◽  
Finite Difference Schemes ◽  
Central Difference ◽  
Unconstrained Optimization Problem ◽  
Optimizing Control ◽  
Necessary Condition Of Optimality

Controlled variable (CV) selection plays an important role in determining the performance of a process plant. Existing methods for CV selection through self-optimizing control requires linearization of rigorous models around nominal operating points. This is a very difficult task which results to large losses. This work presents a novel method for CV design. A necessary condition of optimality (NCO) was proposed to be the CV. The approach does not require the analytical expression of the NCO to be derived but is approximated through a single regression step based on data. Finite difference was used to approximate the NCO (gradient) using data; three finite difference schemes were employed for this purpose, which are forward, backward and central differences. Seven different cases with respect to number of sampling points, neighborhood points and finite difference schemes were investigated. To demonstrate the efficacy of the method in simplest way, it is applied to a hypothetical unconstrained optimization problem. The proposed method was found to have outperformed some existing approaches in many instances. A zero loss was recorded by some designed CVs. Central difference was found to be the best schemes among the three. Keywords— controlled variable, disturbance, finite difference, monotonicity, regression. 

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