scholarly journals Saddle Point Search Algorithm for the Problem of Site Protection Level Assignment Based on Search of Simplices

2019 ◽  
Author(s):  
A.Yu. Bykov ◽  
M.V. Grishunin ◽  
I.A. Krygin
Keyword(s):  
Saddle Point ◽  
Search Algorithm ◽  
Linear Equations ◽  
The Other ◽  
Protection Level ◽  
Point Search ◽  
Systems Of Linear Equations ◽  
Feasible Solutions ◽  
Fixed Solution ◽  
Zero Sum

This paper deals with a continuous zero-sum game with constraints on resources between a defender allocating resources for protection of sites and an attacker choosing sites for attack. The problem is formulated so that each player would have to solve its own linear program with a fixed solution of the other player. We show that in this case the saddle point is located on the faces of simplices defining feasible solutions. We propose an algorithm of saddle point search based on search of the simplices' faces on hyperplanes of equal dimension. Each possible face is defined using a boolean vector defining states of variables and problem constraints. The search of faces is reduced to the search of feasible boolean vectors. In order to reduce computational complexity of the search we formulate the rules for removing patently unfeasible faces. Each point of a face belonging to an (m--1)-dimensional hyperplane is defined using m points of the hyperplane. We created an algorithm for generating these points. Two systems of linear equations must be solved in order to find the saddle point if it located on the faces of simplices belonging to hyperplanes of equal dimension. We created a generic algorithm of saddle point search on the faces located on hyperplanes of equal dimension. We present an example of solving a problem and the results of computational experiments

On one Saddle Point Search Algorithm for Continuous Linear Games as Applied to Information Security Problems

2020 ◽  
pp. 58-74
Author(s):  
A.Yu. Bykov ◽  
I.A. Krygin ◽  
M.V. Grishunin ◽  
I.А. Markova
Keyword(s):  
Saddle Point ◽  
Relative Error ◽  
Optimization Problems ◽  
Search Algorithm ◽  
The Other ◽  
Matrix Games ◽  
Random Number Generators ◽  
Point Search ◽  
Fixed Solution ◽  
Zero Sum

The paper introduces a game formulation of the problem of two players: the defender determines the security levels of objects, and the attacker determines the objects for attack. Each of them distributes his resources between the objects. The assessment of a possible damage to the defender serves as an indicator of quality. The problem of a continuous zero-sum game under constraints on the resources of the players is formulated so that each player must solve his own linear programming problem with a fixed solution of the other player. The purpose of this research was to develop an algorithm for finding a saddle point. The algorithm is approximate and based on reducing a continuous problem to discrete or matrix games of high dimension, since the optimal solutions are located at the vertices or on the faces of the simplices which determine the sets of players' admissible solutions, and the number of vertices or faces of the simplices is finite. In the proposed algorithm, the optimization problems of the players are sequentially solved with the accumulated averaged solution of the other player, in fact, the ideas of the Brown --- Robinson method are used. An example of solving the problem is also given. The paper studies the dependences of the number of algorithm steps on the relative error of the quality indicator and on the dimension of the problem, i.e., the number of protected objects, for a given relative error. The initial data are generated using pseudo-random number generators

Unlocking the Mystery of the Origins of John von Neumann’s Growth Model

2021 ◽  
Vol 53 (4) ◽  
pp. 595-631
Author(s):  
Juan Carvajalino
Keyword(s):  
Linear Equations ◽  
Minimax Theorem ◽  
Theoretical Models ◽  
Von Neumann ◽  
Systems Of Linear Equations ◽  
National Boundaries ◽  
And Mathematics ◽  
Zero Sum ◽  
General Economic Equilibrium ◽  
Local Perspectives

In his famous “A Model of General Economic Equilibrium,” von Neumann wrote that it was “obvious to what kind of theoretical models [his] assumptions correspond.” To date, however, his sources of economic insights about the functioning of the continuously growing price-economy that he modeled have remained a total mystery. Based on archival material, this mystery is solved in this account by making visible the specific influences from economics and mathematics that inspired him. I argue that von Neumann’s 1937 paper resulted from a deep engagement with economics as it was emerging at the beginning of the 1930s and that this happened as he was travelling and crossing national boundaries while bridging distinct branches of mathematics with different local perspectives in economics. His encounters with Jacob Marschak in Berlin, Nicolas Kaldor in Budapest, and Frank Graham in Princeton as well as his reading of Walras’s, Wicksell’s and Cassel’s work would be key. I also explain how he came to realize that there existed a formal analogy between systems of linear equations and inequalities with which he characterized (stationary and dynamic) economies and the minimax theorem for two-person zero-sum games that he had conceived and proved in 1928.

An Efficient Saddle Point Search Method Using Kriging Metamodels

2015 ◽  
Author(s):  
Lijuan He ◽  
Yan Wang
Keyword(s):  
Potential Energy ◽  
Saddle Point ◽  
Density Functional ◽  
Search Algorithm ◽  
Saddle Points ◽  
Surrogate Models ◽  
Functional Theory ◽  
Search Results ◽  
Point Search ◽  
Large Systems

Simulating phase transformation of materials at the atomistic scale requires the knowledge of saddle points on the potential energy surface (PES). In the existing first-principles saddle point search methods, the requirement of a large number of expensive evaluations of potential energy, e.g. using density functional theory (DFT), limits the application of such algorithms to large systems. Thus, it is meaningful to minimize the number of functional evaluations as DFT simulations during the search process. Furthermore, model-form uncertainty and numerical errors are inherent in DFT and search algorithms. Robustness of the search results should be considered. In this paper, a new search algorithm based on Kriging is presented to search local minima and saddle points on a PES efficiently and robustly. Different from existing searching methods, the algorithm keeps a memory of searching history by constructing surrogate models and uses the search results on the surrogate models to provide the guidance of future search on the PES. The surrogate model is also updated with more DFT simulation results. The algorithm is demonstrated by the examples of Rastrigin and Schwefel functions with a multitude of minima and saddle points.

On Some Properties of Semi-rings

2018 ◽  
pp. 35-50
Author(s):  
A. I. Belousov
Keyword(s):  
Linear Equations ◽  
Oriented Graph ◽  
Theoretical Computer Science ◽  
Alternative Methods ◽  
Main Role ◽  
Cost Matrix ◽  
Sequential Elimination ◽  
The Matrix ◽  
Systems Of Linear Equations ◽  
The Cost

The main objective of this paper is to prove a theorem according to which a method of successive elimination of unknowns in the solution of systems of linear equations in the semi-rings with iteration gives the really smallest solution of the system. The proof is based on the graph interpretation of the system and establishes a relationship between the method of sequential elimination of unknowns and the method for calculating a cost matrix of a labeled oriented graph using the method of sequential calculation of cost matrices following the paths of increasing ranks. Along with that, and in terms of preparing for the proof of the main theorem, we consider the following important properties of the closed semi-rings and semi-rings with iteration.We prove the properties of an infinite sum (a supremum of the sequence in natural ordering of an idempotent semi-ring). In particular, the proof of the continuity of the addition operation is much simpler than in the known issues, which is the basis for the well-known algorithm for solving a linear equation in a semi-ring with iteration.Next, we prove a theorem on the closeness of semi-rings with iteration with respect to solutions of the systems of linear equations. We also give a detailed proof of the theorem of the cost matrix of an oriented graph labeled above a semi-ring as an iteration of the matrix of arc labels.The concept of an automaton over a semi-ring is introduced, which, unlike the usual labeled oriented graph, has a distinguished "final" vertex with a zero out-degree.All of the foregoing provides a basis for the proof of the main theorem, in which the concept of an automaton over a semi-ring plays the main role.The article's results are scientifically and methodologically valuable. The proposed proof of the main theorem allows us to relate two alternative methods for calculating the cost matrix of a labeled oriented graph, and the proposed proofs of already known statements can be useful in presenting the elements of the theory of semi-rings that plays an important role in mathematical studies of students majoring in software technologies and theoretical computer science.

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