scholarly journals Computing mixed strategies equilibria in presence of switching costs by the solution of nonconvex QP problems

2021 ◽  
Author(s):  
G. Liuzzi ◽  
M. Locatelli ◽  
V. Piccialli ◽  
S. Rass
Keyword(s):  
Game Theory ◽  
Branch And Bound ◽  
Switching Costs ◽  
Computational Cost ◽  
Linear Constraints ◽  
Point Of View ◽  
Quadratic Term ◽  
Mixed Strategies ◽  
Computational Point ◽  
Spatial Branch And Bound

AbstractIn this paper we address game theory problems arising in the context of network security. In traditional game theory problems, given a defender and an attacker, one searches for mixed strategies which minimize a linear payoff functional. In the problems addressed in this paper an additional quadratic term is added to the minimization problem. Such term represents switching costs, i.e., the costs for the defender of switching from a given strategy to another one at successive rounds of a Nash game. The resulting problems are nonconvex QP ones with linear constraints and turn out to be very challenging. We will show that the most recent approaches for the minimization of nonconvex QP functions over polytopes, including commercial solvers such as and , are unable to solve to optimality even test instances with $$n=50$$ n = 50 variables. For this reason, we propose to extend with them the current benchmark set of test instances for QP problems. We also present a spatial branch-and-bound approach for the solution of these problems, where a predominant role is played by an optimality-based domain reduction, with multiple solutions of LP problems at each node of the branch-and-bound tree. Of course, domain reductions are standard tools in spatial branch-and-bound approaches. However, our contribution lies in the observation that, from the computational point of view, a rather aggressive application of these tools appears to be the best way to tackle the proposed instances. Indeed, according to our experiments, while they make the computational cost per node high, this is largely compensated by the rather slow growth of the number of nodes in the branch-and-bound tree, so that the proposed approach strongly outperforms the existing solvers for QP problems.

scholarly journals A New Spatial Branch and Bound Algorithm for Quadratic Program with One Quadratic Constraint and Linear Constraints

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Jing Zhou
Keyword(s):  
Branch And Bound ◽  
High Efficiency ◽  
Real Life ◽  
Linear Constraints ◽  
Branch And Bound Algorithm ◽  
Quadratic Program ◽  
Constraint Matrix ◽  
Sdp Relaxation ◽  
Quadratic Constraint ◽  
Spatial Branch And Bound

This paper proposes a novel second-order cone programming (SOCP) relaxation for a quadratic program with one quadratic constraint and several linear constraints (QCQP) that arises in various real-life fields. This new SOCP relaxation fully exploits the simultaneous matrix diagonalization technique which has become an attractive tool in the area of quadratic programming in the literature. We first demonstrate that the new SOCP relaxation is as tight as the semidefinite programming (SDP) relaxation for the QCQP when the objective matrix and constraint matrix are simultaneously diagonalizable. We further derive a spatial branch-and-bound algorithm based on the new SOCP relaxation in order to obtain the global optimal solution. Extensive numerical experiments are conducted between the new SOCP relaxation-based branch-and-bound algorithm and the SDP relaxation-based branch-and-bound algorithm. The computational results illustrate that the new SOCP relaxation achieves a good balance between the bound quality and computational efficiency and thus leads to a high-efficiency global algorithm.

COMBINATORIAL POLYNOMIALLY COMPUTABLE CHARACTERISTICS OF SUBSTITUTIONS AND THEIR PROPERTIES

2020 ◽  
Vol 7 (2) ◽  
pp. 34-41
Author(s):  
VLADIMIR NIKONOV ◽  
◽  
ANTON ZOBOV ◽  
Keyword(s):  
Mathematical Expectation ◽  
Point Of View ◽  
Small Range ◽  
Computational Point ◽  
Wide Range ◽  
Bijective Function ◽  
The Difference ◽  
Element Base ◽  
Selection Of

The construction and selection of a suitable bijective function, that is, substitution, is now becoming an important applied task, particularly for building block encryption systems. Many articles have suggested using different approaches to determining the quality of substitution, but most of them are highly computationally complex. The solution of this problem will significantly expand the range of methods for constructing and analyzing scheme in information protection systems. The purpose of research is to find easily measurable characteristics of substitutions, allowing to evaluate their quality, and also measures of the proximity of a particular substitutions to a random one, or its distance from it. For this purpose, several characteristics were proposed in this work: difference and polynomial, and their mathematical expectation was found, as well as variance for the difference characteristic. This allows us to make a conclusion about its quality by comparing the result of calculating the characteristic for a particular substitution with the calculated mathematical expectation. From a computational point of view, the thesises of the article are of exceptional interest due to the simplicity of the algorithm for quantifying the quality of bijective function substitutions. By its nature, the operation of calculating the difference characteristic carries out a simple summation of integer terms in a fixed and small range. Such an operation, both in the modern and in the prospective element base, is embedded in the logic of a wide range of functional elements, especially when implementing computational actions in the optical range, or on other carriers related to the field of nanotechnology.

scholarly journals TOPOLOGICAL INSULATOR AND THE DIRAC EQUATION

SPIN ◽  
2011 ◽  
Vol 01 (01) ◽  
pp. 33-44 ◽  
Author(s):  
SHUN-QING SHEN ◽  
WEN-YU SHAN ◽  
HAI-ZHOU LU
Keyword(s):  
Dirac Equation ◽  
Topological Insulator ◽  
Bound States ◽  
Topological Insulators ◽  
Mass Term ◽  
Point Of View ◽  
Quadratic Term ◽  
General Description ◽  
Dirac Equations ◽  
Topological Quantum

We present a general description of topological insulators from the point of view of Dirac equations. The Z2 index for the Dirac equation is always zero, and thus the Dirac equation is topologically trivial. After the quadratic term in momentum is introduced to correct the mass term m or the band gap of the Dirac equation, i.e., m → m − Bp2, the Z2 index is modified as 1 for mB > 0 and 0 for mB < 0. For a fixed B there exists a topological quantum phase transition from a topologically trivial system to a nontrivial system when the sign of mass m changes. A series of solutions near the boundary in the modified Dirac equation is obtained, which is characteristic of topological insulator. From the solutions of the bound states and the Z2 index we establish a relation between the Dirac equation and topological insulators.

PORTFOLIO MANAGEMENT OF ENERGY SAVING PROJECTS BASED ON THE MARKOVITS THEORY

2021 ◽  
pp. 79-91
Author(s):  
S. Kiyko ◽  
L. Deineha ◽  
M. Basanets ◽  
D. Kamienskyi ◽  
A. Didenko
Keyword(s):  
Energy Saving ◽  
Portfolio Optimization ◽  
Portfolio Management ◽  
Quadratic Function ◽  
Subject Area ◽  
Linear Constraints ◽  
Point Of View ◽  
Investment Portfolio ◽  
Financial Investment ◽  
Software Application

The goal of the work was to identify research and compare methods of portfolio management of energy saving projects and to develop software for optimizing portfolio investments using several methods. The key elements and strategies of creating an effective investment portfolio are considered: diversification, rebalancing, active portfolio management, passive portfolio management. Given the basic principles of investment theory, the task of portfolio investment is to form an investment portfolio with known shares of certain assets to maximize returns and minimize risk. To solve this problem, the method of Harry Markowitz, known as modern portfolio theory, was chosen. This is the theory of financial investment, in which statistical methods are used to make the most profitable risk distribution of the securities portfolio and income valuation, its components are asset valuation, investment decisions, portfolio optimization, evaluation of results. From a mathematical point of view, the problem of forming an optimal portfolio is the problem of optimizing a quadratic function (finding the minimum) with linear constraints on the arguments of the function. Methods of optimization of portfolios of energy saving projects taking into account the specifics of the subject area are analyzed. According to the results of the analysis, the methods of finding the maximum Sharpe’s ratio and the minimum volatility from randomly generated portfolios were chosen. A software application has been developed that allows you to download data, generate random portfolios and optimize them with selected methods. A graphical display of portfolio optimization results has also been implemented. The program was tested on data on shares of energy saving companies. The graphs built by the program allow the operator to better assess the created portfolio of the energy saving project.

scholarly journals The Regularized Weak Functional Matching Pursuit for linear inverse problems

2019 ◽  
Vol 27 (3) ◽  
pp. 317-340 ◽  
Author(s):  
Max Kontak ◽  
Volker Michel
Keyword(s):  
Inverse Problems ◽  
Matching Pursuit ◽  
A Priori ◽  
Computation Time ◽  
Point Of View ◽  
Computational Point ◽  
Linear Inverse Problems ◽  
Infinite Dimensional ◽  
Frame Condition ◽  
Ill Posed

Abstract In this work, we present the so-called Regularized Weak Functional Matching Pursuit (RWFMP) algorithm, which is a weak greedy algorithm for linear ill-posed inverse problems. In comparison to the Regularized Functional Matching Pursuit (RFMP), on which it is based, the RWFMP possesses an improved theoretical analysis including the guaranteed existence of the iterates, the convergence of the algorithm for inverse problems in infinite-dimensional Hilbert spaces, and a convergence rate, which is also valid for the particular case of the RFMP. Another improvement is the cancellation of the previously required and difficult to verify semi-frame condition. Furthermore, we provide an a-priori parameter choice rule for the RWFMP, which yields a convergent regularization. Finally, we will give a numerical example, which shows that the “weak” approach is also beneficial from the computational point of view. By applying an improved search strategy in the algorithm, which is motivated by the weak approach, we can save up to 90  of computation time in comparison to the RFMP, whereas the accuracy of the solution does not change as much.

Computationally Efficient Simulation of Multi-Component Fuel Combustion Using a Sparse Analytical Jacobian Chemistry Solver and High-Dimensional Clustering

2013 ◽  
Author(s):  
Federico Perini ◽  
Anand Krishnasamy ◽  
Youngchul Ra ◽  
Rolf D. Reitz
Keyword(s):  
Reaction Mechanism ◽  
Chemical Kinetics ◽  
Internal Combustion Engines ◽  
Combustion Engine ◽  
Engine Performance ◽  
Internal Combustion ◽  
Point Of View ◽  
High Dimensional ◽  
Computational Point ◽  
Fuel Surrogates

The need for more efficient and environmentally sustainable internal combustion engines is driving research towards the need to consider more realistic models for both fuel physics and chemistry. As far as compression ignition engines are concerned, phenomenological or lumped fuel models are unreliable to capture spray and combustion strategies outside of their validation domains — typically, high-pressure injection and high-temperature combustion. Furthermore, the development of variable-reactivity combustion strategies also creates the need to model comprehensively different hydrocarbon families even in single fuel surrogates. From the computational point of view, challenges to achieving practical simulation times arise from the dimensions of the reaction mechanism, that can be of hundreds species even if hydrocarbon families are lumped into representative compounds, and thus modeled with non-elementary, skeletal reaction pathways. In this case, it is also impossible to pursue further mechanism reductions to lower dimensions. CPU times for integrating chemical kinetics in internal combustion engine simulations ultimately scale with the number of cells in the grid, and with the cube number of species in the reaction mechanism. In the present work, two approaches to reduce the demands of engine simulations with detailed chemistry are presented. The first one addresses the demands due to the solution of the chemistry ODE system, and features the adoption of SpeedCHEM, a newly developed chemistry package that solves chemical kinetics using sparse analytical Jacobians. The second one aims to reduce the number of chemistry calculations by binning the CFD cells of the engine grid into a subset of clusters, where chemistry is solved and then mapped back to the original domain. In particular, a high-dimensional representation of the chemical state space is adopted for keeping track of the different fuel components, and a newly developed bounding-box-constrained k-means algorithm is used to subdivide the cells into reactively homogeneous clusters. The approaches have been tested on a number of simulations featuring multi-component diesel fuel surrogates, and different engine grids. The results show that significant CPU time reductions, of about one order of magnitude, can be achieved without loss of accuracy in both engine performance and emissions predictions, prompting for their applicability to more refined or full-sized engine grids.

scholarly journals Controllability and Observability of a Large Scale Thermodynamical System via Connectability Approach

2010 ◽  
Author(s):  
Virdiansyah Permana ◽  
Rahmat Shoureshi
Keyword(s):  
Graph Theory ◽  
Lie Algebra ◽  
Large Scale ◽  
Point Of View ◽  
Rank Condition ◽  
Computational Point ◽  
New Approach ◽  
Class System ◽  
Necessary And Sufficient ◽  
Controllability And Observability

This study presents a new approach to determine the controllability and observability of a large scale nonlinear dynamic thermal system using graph-theory. The novelty of this method is in adapting graph theory for nonlinear class and establishing a graphic condition that describes the necessary and sufficient terms for a nonlinear class system to be controllable and observable, which equivalents to the analytical method of Lie algebra rank condition. The directed graph (digraph) is utilized to model the system, and the rule of its adaptation in nonlinear class is defined. Subsequently, necessary and sufficient terms to achieve controllability and observability condition are investigated through the structural property of a digraph called connectability. It will be shown that the connectability condition between input and states, as well as output and states of a nonlinear system are equivalent to Lie-algebra rank condition (LARC). This approach has been proven to be easier from a computational point of view and is thus found to be useful when dealing with a large system.

MONTE CARLO METHODS IN FUZZY GAME THEORY

2007 ◽  
Vol 03 (02) ◽  
pp. 259-269 ◽  
Author(s):  
AREEG ABDALLA ◽  
JAMES BUCKLEY
Keyword(s):  
Game Theory ◽  
Monte Carlo ◽  
Monte Carlo Method ◽  
Monte Carlo Methods ◽  
Mixed Strategy ◽  
Minimax Theorem ◽  
Approximate Solutions ◽  
Mixed Strategies ◽  
Fuzzy Game ◽  
Zero Sum

In this paper, we consider a two-person zero-sum game with fuzzy payoffs and fuzzy mixed strategies for both players. We define the fuzzy value of the game for both players [Formula: see text] and also define an optimal fuzzy mixed strategy for both players. We then employ our fuzzy Monte Carlo method to produce approximate solutions, to an example fuzzy game, for the fuzzy values [Formula: see text] for Player I and [Formula: see text] for Player II; and also approximate solutions for the optimal fuzzy mixed strategies for both players. We then look at [Formula: see text] and [Formula: see text] to see if there is a Minimax theorem [Formula: see text] for this fuzzy game.

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