ON COMPUTATIONAL COMPLEXITY OF HIERARCHICAL OPTIMIZATION

2002 ◽  
Vol 13 (05) ◽  
pp. 667-670
Author(s):  
WEIJIA JIA ◽  
ZHIBIN SUN
Keyword(s):  
Nash Equilibrium ◽  
Computational Complexity ◽  
Objective Function ◽  
Optimization Problem ◽  
Linear Constraints ◽  
The Other ◽  
Hierarchical Optimization ◽  
Linear Objective Function ◽  
Decision Variables

In this work, the computational complexity of a hierarchic optimization problem involving in several players is studied. Each player is assigned with a linear objective function. The set of variables is partitioned such that each subset corresponds to one player as its decision variables. All the players jointly make a decision on the values of these variables such that a set of linear constraints should be satisfied. One special player, called the leader, makes decision on its decision variables before of all the other players. The rest, after learnt of the decision of the leader, make their choices so that their decisions form a Nash Equilibrium for them, breaking tie by maximizing the objective function of player. We show that the exact complexity of the problem is FPNP-complete.

scholarly journals Maximin and Minimax Strategies in Two-Players Game with Two Strategic Variables

2018 ◽  
Vol 20 (01) ◽  
pp. 1750030 ◽  
Author(s):  
Atsuhiro Satoh ◽  
Yasuhito Tanaka
Keyword(s):  
Nash Equilibrium ◽  
Objective Function ◽  
The Other ◽  
Minimax Strategy ◽  
Equilibrium Strategies ◽  
Minimax Strategies ◽  
Nash Equilibrium Strategies ◽  
Special Case ◽  
Maximin Strategy

We examine maximin and minimax strategies for players in a two-players game with two strategic variables, [Formula: see text] and [Formula: see text]. We consider two patterns of game; one is the [Formula: see text]-game in which the strategic variables of players are [Formula: see text]’s, and the other is the [Formula: see text]-game in which the strategic variables of players are [Formula: see text]’s. We call two players Players A and B, and will show that the maximin strategy and the minimax strategy in the [Formula: see text]-game, and the maximin strategy and the minimax strategy in the [Formula: see text]-game are all equivalent for each player. However, the maximin strategy for Player A and that for Player B are not necessarily equivalent, and they are not necessarily equivalent to their Nash equilibrium strategies in the [Formula: see text]-game nor the [Formula: see text]-game. But, in a special case, where the objective function of Player B is the opposite of the objective function of Player A, the maximin strategy for Player A and that for Player B are equivalent, and they constitute the Nash equilibrium both in the [Formula: see text]-game and the [Formula: see text]-game.

Research on Precise Constraint Reconstruction of Section Data

2013 ◽  
Vol 712-715 ◽  
pp. 1122-1125
Author(s):  
Xu Zhang ◽  
Hai Bo Zhang ◽  
Xue Chang Zhang
Keyword(s):  
Objective Function ◽  
Optimization Problem ◽  
Linear Constraints ◽  
Reconstruction Method ◽  
Linear Boundary ◽  
Reconstruction Accuracy ◽  
B Spline ◽  
Boundary Constraints ◽  
Real Application ◽  
Section Data

Since the features are not prominent and the algorithm is complex during whole reconstruction of section data, a step-by-step optimization reconstruction method is proposed. The order of reconstruction is optimized: line features are reconstructed firstly; Then arc features are reconstructed (constraints are satisfied between the arc and known lines); finally freeform features are reconstructed (constraints are satisfied between the B-spline and known lines/arcs). In this way, the reconstruction accuracy of the line/arc features is ensured in the first. Since freeform features have more freedom, it is convenient to be adjusted to meet the more constraints. Linear boundary constraints are constructed and the algorithm becomes optimization problem of the quadratic objective function under the linear constraints. The examples show that the reconstruction accuracy is improved greatly under satisfying constraints; the expected goal is achieved in real application.

Transient Optimization of Natural Gas Networks Using Intelligent Algorithms

2018 ◽  
Vol 141 (3) ◽  
Author(s):  
Esmaeel Khanmirza ◽  
Reza Madoliat ◽  
Ali Pourfard
Keyword(s):  
Natural Gas ◽  
Power Consumption ◽  
Objective Function ◽  
Optimization Problem ◽  
Transient Condition ◽  
Continuous Variables ◽  
Intelligent Algorithms ◽  
Gas Network ◽  
Gas Networks ◽  
Decision Variables

Compressor stations in natural gas networks should perform such that time-varying demands of customers are fulfilled while all of the system constraints are satisfied. Power consumption of compressor stations impose the most operational cost to a gas network so their optimal performance will lead to significant money saving. In this paper, the gas network transient optimization problem is addressed. The objective function is the sum of the compressor's power consumption that should be minimized where compressor speeds and the value status are decision variables. This objective function is nonlinear which is subjected to nonlinear and combinatorial constraints including both discrete and continuous variables. To handle this challenging optimization problem, a novel approach based on using two different structure intelligent algorithms, namely the particle swarm optimization (PSO) and cultural algorithm (CA), is utilized to find the optimum of the decision variables. This approach removes the necessity of finding an explicit expression for the power consumption of compressors as a function of decision variables as well as the calculation of objective function derivatives. The objective function and constraints are evaluated in the transient condition by a fully implicit finite difference numerical method. The proposed approach is applied on a real gas network where simulation results confirm its accuracy and efficiency.

Development of a Spreadsheet DSS for Multi-Response Taguchi Parameter Optimization Problems Using the TOPSIS, VIKOR, and GRA Methods

2019 ◽  
Vol 18 (05) ◽  
pp. 1501-1531 ◽  
Author(s):  
Bariş Keçeci ◽  
Yusuf Tansel Iç ◽  
Ergün Eraslan
Keyword(s):  
Objective Function ◽  
Parameter Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Objective Function Value ◽  
Optimal Parameter ◽  
The Other ◽  
Mixed Integer ◽  
Parameter Combination ◽  
Single Response

This paper presents a spreadsheet-based decision support system (DSS) for any parameter optimization problem, in the small- and medium-sized enterprises to help the managers to make better decisions. Microsoft Excel is used as a DSS development platform. The DSS application requires the quality characteristics and the level of parameters affecting the problem. The proposed system considers three multi-criteria decision-making methods: TOPSIS, VIKOR and GRA. These methods are integrated into the Taguchi method to convert the multi-response optimization problem to a single-response problem. The DSS suggests proper Taguchi experimental designs and provides the decision maker with an opportunity to use different metrics and to validate the experimental results. Several issues and an application are provided for illustrative purposes. The proposed DSS is tested on a case study (the performance of the mixed integer programming (MIP) formulation solver) and the results highlight that the system is capable of offering satisfactory outcomes. Using such a quick and flexible DSS might help to reduce the daily workload of the decision makers. The different metrics used for the response variables which results with the different parameter combination. Using the optimal parameter combination of TOPSIS (come to the fore in case MinBest metric used), the MIP formulation solver gives the best integer objective function value of 609 and a GAP value of 1.93%, both of which are less than the values obtained using the other methods. Using the optimal parameter combination of GRA (come to the fore in case OptBest metric used), the MIP formulation gives a best integer objective function value of 632 and a GAP value of 6.52%, both of which are less than the values obtained by using the other methods.

Convex Grey Optimization

2019 ◽  
Vol 53 (1) ◽  
pp. 339-349
Author(s):  
Surafel Luleseged Tilahun
Keyword(s):  
Objective Function ◽  
Optimization Problem ◽  
Fuzzy Numbers ◽  
Optimization Problems ◽  
Probability Distributions ◽  
Optimal Solution ◽  
Decision Variables ◽  
Constraint Set ◽  
Grey Number ◽  
Real Scenario

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.

scholarly journals Hypergraph Optimization Problems: Why is the Objective Function Linear?

1996 ◽  
Vol 3 (50) ◽  
Author(s):  
Aleksandar Pekec
Keyword(s):  
Objective Function ◽  
Optimization Problem ◽  
Optimization Problems ◽  
A Priori ◽  
Point Of View ◽  
Linear Objective Function ◽  
Discrete Structures

Choosing an objective function for an optimization problem is a<br />modeling issue and there is no a-priori reason that the objective function<br />must be linear. Still, it seems that linear 0-1 programming formulations<br />are overwhelmingly used as models for optimization problems<br />over discrete structures. We show that this is not an accident. Under<br />some reasonable conditions (from the modeling point of view), the<br />linear objective function is the only possible one.

Shape Optimization of Hydraulic Structures: an Example of an Optimum Design of a Fish Passage

10.29007/2k64 ◽  
2018 ◽  
Author(s):  
Pat Prodanovic ◽  
Cedric Goeury ◽  
Fabrice Zaoui ◽  
Riadh Ata ◽  
Jacques Fontaine ◽  
...  
Keyword(s):  
Numerical Model ◽  
Shape Optimization ◽  
Objective Function ◽  
Optimum Design ◽  
Optimization Problem ◽  
A Priori ◽  
Coupled System ◽  
Fish Passage ◽  
Hydraulic Structures ◽  
Design Variables

This paper presents a practical methodology developed for shape optimization studies of hydraulic structures using environmental numerical modelling codes. The methodology starts by defining the optimization problem and identifying relevant problem constraints. Design variables in shape optimization studies are configuration of structures (such as length or spacing of groins, orientation and layout of breakwaters, etc.) whose optimal orientation is not known a priori. The optimization problem is solved numerically by coupling an optimization algorithm to a numerical model. The coupled system is able to define, test and evaluate a multitude of new shapes, which are internally generated and then simulated using a numerical model. The developed methodology is tested using an example of an optimum design of a fish passage, where the design variables are the length and the position of slots. In this paper an objective function is defined where a target is specified and the numerical optimizer is asked to retrieve the target solution. Such a definition of the objective function is used to validate the developed tool chain. This work uses the numerical model TELEMAC- 2Dfrom the TELEMAC-MASCARET suite of numerical solvers for the solution of shallow water equations, coupled with various numerical optimization algorithms available in the literature.

A new algorithm for finding CRM-model coefficients

2019 ◽  
Vol 5 (3) ◽  
pp. 164-185 ◽  
Author(s):  
Alexander D. Bekman ◽  
Sergey V. Stepanov ◽  
Alexander A. Ruchkin ◽  
Dmitry V. Zelenin
Keyword(s):  
Approximate Solution ◽  
Optimization Problem ◽  
Optimal Solution ◽  
Complex Form ◽  
Approximate Solutions ◽  
The Other ◽  
Programming Algorithm ◽  
Stochastic Algorithms ◽  
Large Set ◽  
Flow Rates

The quantitative evaluation of producer and injector well interference based on well operation data (profiles of flow rates/injectivities and bottomhole/reservoir pressures) with the help of CRM (Capacitance-Resistive Models) is an optimization problem with large set of variables and constraints. The analytical solution cannot be found because of the complex form of the objective function for this problem. Attempts to find the solution with stochastic algorithms take unacceptable time and the result may be far from the optimal solution. Besides, the use of universal (commercial) optimizers hides the details of step by step solution from the user, for example&nbsp;— the ambiguity of the solution as the result of data inaccuracy.<br> The present article concerns two variants of CRM problem. The authors present a new algorithm of solving the problems with the help of “General Quadratic Programming Algorithm”. The main advantage of the new algorithm is the greater performance in comparison with the other known algorithms. Its other advantage is the possibility of an ambiguity analysis. This article studies the conditions which guarantee that the first variant of problem has a unique solution, which can be found with the presented algorithm. Another algorithm for finding the approximate solution for the second variant of the problem is also considered. The method of visualization of approximate solutions set is presented. The results of experiments comparing the new algorithm with some previously known are given.

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