Dual Linear Programming Research Articles

An Integrated SIMUS–Game Theory Approach for Sustainable Decision Making—An Application for Route and Transport Operator Selection

The choice of management strategy for companies operating in different sectors of the economy is of great importance for their sustainable development. In many cases, companies are in competition within the scope of the same activities, meaning that the profit of one company is at the expense of the other. The choice of strategies for each of the firms in this case can be optimized using game theory for a non-cooperative game case where the two players have antagonistic interests. The aim of this research is to develop a methodology which, in non-cooperative games, accounts for the benefits of different criteria for each of the strategies of the two participants. In this research a new integrated sequential interactive model for urban systems (SIMUS)–game theory technique for decision making in the case of non-cooperative games is proposed. The methodology includes three steps. The first step consists of a determination of the strategies of both players and the selection of criteria for their assessment. In the second step the SIMUS method for multi-criteria analysis is applied to identify the benefits of the strategies for both players according to the criteria. The model formation in game theory is drawn up in the third step. The payoff matrix of the game is formed based on the benefits obtained from the SIMUS method. The strategies of both players are solved by dual linear programming. Finally, to verify the results of the new approach we apply four criteria to make a decision—Laplace’s criterion, the minimax and maximin criteria, Savage’s criterion and Hurwitz’s criterion. The new integrated SIMUS–game theory approach is applied to a real example in the transport sector. The Bulgarian transport network is investigated regarding route and transport type selection for a carriage of containers between a starting point, Sofia, and a destination, Varna, in the case of competition between railway and road operators. Two strategies for a railway operator and three strategies for a road operator are examined. The benefits of the strategies for both operators are determined using the SIMUS method, based on seven criteria representing environmental, technological, infrastructural, economic, security and safety factors. The optimal strategies for both operators are determined using the game model and dual linear programming. It is discovered that the railway operator will apply their first strategy and that the road operator will also apply their first strategy. Both players will obtain a profit if they implement their optimal strategies. The new integrated SIMUS–game theory approach can be used in different areas of research, when the strategies for both players in non-cooperatives games need to be established.

Max-flow min-cut theorem in quantum computing

The max-flow min-cut theorem in graph theory states that the maximum flow of a network is equal to the minimum cut of edges of the flow network. Max-flow and min-cut are related as two primal–dual linear programs. The theorem is applicable in applications like network connectivity, graph matching problems, transportation and logistics, and scheduling problems. The aim of the paper is to model the classical max-flow and min-cut theorem in quantum computing. Two well-known methods for quantum optimization are quantum annealing and quantum approximate optimization algorithm. The max-flow min-cut is converted to its equivalent model of the above two methods for execution in polynomial time. This paper shows detailed classical to quantum conversion, analysis, implementation, and result of the theorem.

Sparse randomized policies for Markov decision processes based on Tsallis divergence regularization

This work investigates a somewhat different point of view on Markov decision processes by reinterpreting them as a randomized shortest paths problem on a bipartite graph, therefore establishing bridges with entropy-regularized reinforcement learning. The graph structure contains the set of states as “left” nodes and the set of actions as “right” nodes. In that context, the action-to-state transition probabilities are provided by the environment whereas the state-to-action probabilities correspond to the (stochastic) policy to be found. The randomized shortest paths formalism (minimizing expected cost to the goal state subject to (Shannon or Tsallis) relative entropy regularization) is then readily applied to this bipartite structure, providing a possibly sparse stochastic policy interpolating between a least-cost and a purely random policy. The algorithm computing the policy is closely related to the dual linear programming formulation of the Markov decision processes to which the relative entropy regularization term, multiplied by a scaling factor balancing exploitation and exploration (the temperature), is added. It is derived from well-known techniques of discrete optimal control, relying on costates (Lagrange parameters) backward computation. In summary, the proposed algorithm allows the design of optimal stochastic – but still sparse – policies, ranging from a purely rational to a random behavior, depending on the temperature parameter.

Descent along nodal straight lines and simplex algorithm: two variants of regression analysis based on the least absolute deviation method

A comparative analysis of the computational complexity of exact algorithms for estimating linear regression equations was conducted using the least absolute deviation method. The goal of the study is to compare the computational efficiency of exact algorithms for descent along nodal lines and algorithms based on solving linear programming problems. For this purpose, the algorithm of gradient descent along nodal lines and algorithms for solving the equivalent primal and dual linear programming problems using the simplex method were considered. The computational complexity of algorithms for implementing the method of least modules in solving direct and dual linear programming problems was estimated. A comparison between the average time for determining the regression coefficients using the primal and dual linear programming problems and the average time for gradient descent along nodal lines was conducted using the Monte Carlo method of statistical experiments. It is shown that both options are significantly inferior behind gradient descent along nodal lines, both in terms of the computational complexity of the algorithms and in terms of computation time, and this advantage increases with the sample size, reaching hundred times or more.

Efficient Constrained K-center Clustering with Background Knowledge

Center-based clustering has attracted significant research interest from both theory and practice. In many practical applications, input data often contain background knowledge that can be used to improve clustering results. In this work, we build on widely adopted k-center clustering and model its input background knowledge as must-link (ML) and cannot-link (CL) constraint sets. However, most clustering problems including k-center are inherently NP-hard, while the more complex constrained variants are known to suffer severer approximation and computation barriers that significantly limit their applicability. By employing a suite of techniques including reverse dominating sets, linear programming (LP) integral polyhedron, and LP duality, we arrive at the first efficient approximation algorithm for constrained k-center with the best possible ratio of 2. We also construct competitive baseline algorithms and empirically evaluate our approximation algorithm against them on a variety of real datasets. The results validate our theoretical findings and demonstrate the great advantages of our algorithm in terms of clustering cost, clustering quality, and running time.

A Gradient-Aware Search Algorithm for Constrained Markov Decision Processes.

The canonical solution methodology for finite constrained Markov decision processes (CMDPs), where the objective is to maximize the expected infinite-horizon discounted rewards subject to the expected infinite-horizon discounted costs' constraints, is based on convex linear programming (LP). In this brief, we first prove that the optimization objective in the dual linear program of a finite CMDP is a piecewise linear convex (PWLC) function with respect to the Lagrange penalty multipliers. Next, we propose a novel, provably optimal, two-level gradient-aware search (GAS) algorithm which exploits the PWLC structure to find the optimal state-value function and Lagrange penalty multipliers of a finite CMDP. The proposed algorithm is applied in two stochastic control problems with constraints for performance comparison with binary search (BS), Lagrangian primal-dual optimization (PDO), and LP. Compared with the benchmark algorithms, it is shown that the proposed GAS algorithm converges to the optimal solution quickly without any hyperparameter tuning. In addition, the convergence speed of the proposed algorithm is not sensitive to the initialization of the Lagrange multipliers.

A New Minimax Theorem for Randomized Algorithms

The celebrated minimax principle of Yao says that for any Boolean-valued function f with finite domain, there is a distribution μ over the domain of f such that computing f to error ε against inputs from μ is just as hard as computing f to error ε on worst-case inputs. Notably, however, the distribution μ depends on the target error level ε: the hard distribution which is tight for bounded error might be trivial to solve to small bias, and the hard distribution which is tight for a small bias level might be far from tight for bounded error levels. In this work, we introduce a new type of minimax theorem which can provide a hard distribution μ that works for all bias levels at once. We show that this works for randomized query complexity, randomized communication complexity, some randomized circuit models, quantum query and communication complexities, approximate polynomial degree, and approximate logrank. We also prove an improved version of Impagliazzo’s hardcore lemma. Our proofs rely on two innovations over the classical approach of using Von Neumann’s minimax theorem or linear programming duality. First, we use Sion’s minimax theorem to prove a minimax theorem for ratios of bilinear functions representing the cost and score of algorithms. Second, we introduce a new way to analyze low-bias randomized algorithms by viewing them as “forecasting algorithms” evaluated by a certain proper scoring rule. The expected score of the forecasting version of a randomized algorithm appears to be a more fine-grained way of analyzing the bias of the algorithm. We show that such expected scores have many elegant mathematical properties—for example, they can be amplified linearly instead of quadratically. We anticipate forecasting algorithms will find use in future work in which a fine-grained analysis of small-bias algorithms is required.

Improvement of the enterprise’s production program as a way to adapt to market changes

Purpose. Proving the possibility of improving the procedure to adapt the enterprise to market trends based on the self adjustment of its production program using the solutions of direct and dual linear programming problems with the help of the Microsoft Excel package “Search for solutions” add-on. Methodology. The methodological basis of the study is the provisions of modern economic theory, the fundamental works by foreign and domestic scientists on the formation of management systems and the peculiarities of enterprises’ adaptation to changes. The methodological apparatus of economic and mathematical modeling, operations research, in particular, the theory of duality of linear programming problems is widely used. Findings. The enterprise’s adaptation to changes in the external and internal environment with the help of the proposed iterative procedure of unused reserves redistribution between scarce resources made it possible to increase the products’ sale by reducing the manufacturing of one type of products and increasing the output of another one under the conditions of a stable range of products manufacturing at the enterprise. Originality. The article substantiates the procedure of the enterprise adaptation to changes in internal and external environments on the basis of self adjustment of its production program using the solutions of direct and dual problems of linear programming. Practical value. The article features the applied aspects to self adjust the enterprise production program, aimed to increase the efficiency of economic activity by means of step-by-step improvement of the optimal production plan (works, services).

Randomized Strategies for Robust Combinatorial Optimization with Approximate Separation

In this paper, we study the following robust optimization problem. Given a set family representing feasibility and candidate objective functions, we choose a feasible set, and then an adversary chooses one objective function, knowing our choice. The goal is to find a randomized strategy (i.e., a probability distribution over the feasible sets) that maximizes the expected objective value in the worst case. This problem is fundamental in wide areas such as artificial intelligence, machine learning, game theory, and optimization. To solve the problem, we provide a general framework based on the dual linear programming problem. In the framework, we utilize the ellipsoid algorithm with the approximate separation algorithm. We prove that there exists an α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-approximation algorithm for our robust optimization problem if there exists an α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-approximation algorithm for finding a (deterministic) feasible set that maximizes a nonnegative linear combination of the candidate objective functions. Using our result, we provide approximation algorithms for the max–min fair randomized allocation problem and the maximum cardinality robustness problem with a knapsack constraint.

How a Game Theoretic Approach Can Minimize the Cost of Train Passenger Services: An Intermodal Competition between Rail and Road Transport

Game theory models provide very powerful tools for evaluating strategies that are beneficial to both rail and road operators competing for passengers on parallel routes. This study examines how game theory can help rail operators who are incurring losses on passenger transport to identify strategies that can minimise costs, using the methodology of dual linear programming to analyse strategies. In identifying the best strategies for minimising costs for the railway operator, the best strategies for maximising profits for the road operators are also identified. The game model is set up between two passenger transport operators (rail and road) and is based on the income earned by the road operators from passengers. This study illustrates the following: how the strategies of the two competitors (rail and road) are determined; the formation of the payoff matrix and the presentation of the mathematical problem for the two competitors; and the results and verification of the best strategies for both competitors. The Leonid Hurwicz criterion was used to verify the optimal strategies.

Zero-Sum Games and Linear Programming Duality

The minimax theorem for zero-sum games is easily proved from the strong duality theorem of linear programming. For the converse direction, the standard proof by Dantzig is known to be incomplete. We explain and combine classical theorems about solving linear equations with nonnegative variables to give a correct alternative proof more directly than Adler. We also extend Dantzig’s game so that any max-min strategy gives either an optimal LP solution or shows that none exists.

Research on Matching and Vertex Cover Problems in Bipartite Graphs using Simplex Method

This paper considers a bipartite graph where a perfect matching does not necessarily exist. Linear programming is used in this paper as a special case of the linear program is the assignment problem, which is another name for the weighted maximum matching problem. The objective is to show that linear programming, in particular the simplex algorithm, can be used to calculate maximum weight matchings and minimum weighted vertex covers. This study also calculates and shows the equivalence of the maximum matching and minimum vertex cover cardinalities, and uses linear programming duality to present its relevance to König's theorem. Finally, Hall’s marriage theorem is used to explicitly prove the absence of a perfect matching in particular graphs. There exist many algorithms developed over the years that are designed to solve maximum matching problems in polynomial time, but the simplex method is one of the oldest and simplest algorithms to understand in solving these problems.

Strong duality in parametric robust semi-definite linear programming and exact relaxations

This paper addresses the issue of which strong duality holds between parametric robust semi-definite linear optimization problems and their dual programs. In the case of a spectral norm uncertainty set, it yields a corresponding strong duality result with a semi-definite programming as its dual. We also show that the dual can be reformulated as a second-order cone programming problem or a linear programming problem when the constraint uncertainty sets of parametric robust semi-definite linear programs are given in terms of affinely parameterized diagonal matrix.

Growth of Replacements

The following game in a similar formulation to Petri nets and chip-firing games is studied: Given a finite collection of baskets, each has an infinite number of balls of the same value. Initially, a ball from some basket is chosen to put on the table. Subsequently, in each step a ball from the table is chosen to be replaced by some $2$ balls from some baskets. Which baskets to take depend only on the ball to be replaced and they are decided in advance. Given some $n$, the object of the game is to find the maximum possible sum of values $g(n)$ for a table of $n$ balls.
 In this article, the sequence $g(n)/n$ for $n=1,2,\dots$ will be shown to converge to a growth rate $\lambda$. Furthermore, this value $\lambda$ is also the rate of a structure called pseudo-loop and the solution of a rather simple linear program. The structure and the linear program are closely related, e.g. a solution of the linear program gives a pseudo-loop with the rate $\lambda$ in linear time of the number of baskets, and vice versa with the pseudo-loop giving a solution to the dual linear program. A method to test in quadratic time whether a given $\lambda_0$ is smaller than $\lambda$ is provided to approximate $\lambda$. When the values of the balls are all rational, we can compute the precise value of $\lambda$ in cubic time, using the quadratic time rate test algorithm and the binary search with a special condition to stop. Four proofs of the limit $\lambda$ are given: one just uses the relation between the baskets, one uses pseudo-loops, one uses the linear program and one uses Fekete's lemma (the latest proof assumes a condition on the rule of replacements).

Finding a Shortest Non-Zero Path in Group-Labeled Graphs

We study a constrained shortest path problem in group-labeled graphs with nonnegative edge length, called the shortest non-zero path problem. Depending on the group in question, this problem includes two types of tractable variants in undirected graphs: one is the parity-constrained shortest path/cycle problem, and the other is computing a shortest noncontractible cycle in surface-embedded graphs. For the shortest non-zero path problem with respect to finite abelian groups, Kobayashi and Toyooka (2017) proposed a randomized, pseudopolynomial-time algorithm via permanent computation. For a slightly more general class of groups, Yamaguchi (2016) showed a reduction of the problem to the weighted linear matroid parity problem. In particular, some cases are solved in strongly polynomial time via the reduction with the aid of a deterministic, polynomial-time algorithm for the weighted linear matroid parity problem developed by Iwata and Kobayashi (2021), which generalizes a well-known fact that the parity-constrained shortest path problem is solved via weighted matching. In this paper, as the first general solution independent of the group, we present a rather simple, deterministic, and strongly polynomial-time algorithm for the shortest non-zero path problem. This result captures a common tractable feature behind the parity and topological constraints in the shortest path/cycle problem. The algorithm is based on Dijkstra’s algorithm for the unconstrained shortest path problem and Edmonds’ blossom shrinking technique in matching algorithms; this approach is inspired by Derigs’ faster algorithm (1985) for the parity-constrained shortest path problem via a reduction to weighted matching. Furthermore, we improve our algorithm so that it does not require explicit blossom shrinking, and make the computational time match Derigs’ one. In the speeding-up step, a dual linear programming formulation of the equivalent problem based on potential maximization for the unconstrained shortest path problem plays a key role.

Maximum diameter of 3‐ and 4‐colorable graphs

AbstractErdős et al. made conjectures for the maximum diameter of connected graphs without a complete subgraph , which have order and minimum degree . Settling a weaker version of a problem, by strengthening the ‐free condition to ‐colorable, we solve the problem for and using a unified linear programming duality approach. The case is a substantial simplification of the result of Czabarka et al.

The n-queens completion problem

An n-queens configuration is a placement of n mutually non-attacking queens on an ntimes n chessboard. The n-queens completion problem, introduced by Nauck in 1850, is to decide whether a given partial configuration can be completed to an n-queens configuration. In this paper, we study an extremal aspect of this question, namely: how small must a partial configuration be so that a completion is always possible? We show that any placement of at most n/60 mutually non-attacking queens can be completed. We also provide partial configurations of roughly n/4 queens that cannot be completed and formulate a number of interesting problems. Our proofs connect the queens problem to rainbow matchings in bipartite graphs and use probabilistic arguments together with linear programming duality.

SIR Epidemics with State-Dependent Costs and ICU Constraints: A Hamilton–Jacobi Verification Argument and Dual LP Algorithms

The aim of this paper is twofold. On one hand, we strive to give a simpler proof of the optimality of greedy controls when the cost of interventions is control-affine and the dynamics follow a state-constrained controlled SIR model. This is achieved using the Hamilton–Jacobi characterization of the value function, via the verification argument and explicit trajectory-based computations. Aside from providing an alternative to the Pontryagin complex arguments in Avram et al. (Appl Math Comput 418:126816, 2022) (see also Avram et al. in Appl Math Comput 423:127012, 2022), this method allows one to consider more general classes of costs; in particular state-dependent ones. On the other hand, the paper is completed by linear programming methods allowing one to deal with possibly discontinuous costs. In particular, we propose a brief exposition of classes of linearized dynamic programming principles based on our previous work and ensuing dual linear programming algorithms. We emphasize the particularities of our state space and possible generations of forward scenarios using the description of reachable sets.

A novel Lower Bound Limit Analysis model with hexahedron elements for the failure analysis of laboratory and thin infill masonry walls in two-way bending

A novel Lower Bound (LB) Limit Analysis (LA) Finite Element (FE) model for the study at failure of laboratory and thin infill masonry walls in two-way bending is presented. In the model, a masonry plate is discretized into infinitely resistant hexahedrons and quadrilateral interfaces where all plastic dissipation occurs. Three internal static variables act on interfaces, namely bending moment, out-of-plane Kirchhoff shear and torque. Equilibrium is imposed on hexahedrons, whereas admissibility is enforced exclusively on interfaces between adjoining elements. The admissibility of the internal actions on masonry interfaces is imposed with a strength domain obtained by means of an already existing LB homogenization technique where joints are reduced to interfaces. The resultant Linear Programming (LP) problem - which allows to estimate collapse loads and distribution of internal actions at collapse - is characterized by a number of variables and constraints relatively limited, which requires little computational burden to be solved through standard interior point LP software. Failure mechanisms are obtained solving the dual LP problem. The procedure is validated against two existing experimental datasets of walls in two-way bending, namely three series of panels with and without perforations tested at collapse at the University of Adelaide Australia and four series of solid and perforated panels tested at the University of Plymouth UK. Comparing the results in terms of collapse loads, active failure mechanisms and distribution of internal actions with both experimental evidence and previously presented numerical models, excellent results are found, an outcome showing the reliability of the procedure proposed.

Limit laws for empirical optimal solutions in random linear programs

We consider a general linear program in standard form whose right-hand side constraint vector is subject to random perturbations. For the corresponding random linear program, we characterize under general assumptions the random fluctuations of the empirical optimal solutions around their population quantities after standardization by a distributional limit theorem. Our approach is geometric in nature and further relies on duality and the collection of dual feasible basic solutions. The limiting random variables are driven by the amount of degeneracy inherent in linear programming. In particular, if the corresponding dual linear program is degenerate the asymptotic limit law might not be unique and is determined from the way the empirical optimal solution is chosen. Furthermore, we include consistency and convergence rates of the Hausdorff distance between the empirical and the true optimality sets as well as a limit law for the empirical optimal value involving the set of all dual optimal basic solutions. Our analysis is motivated from statistical optimal transport that is of particular interest here and distributional limit laws for empirical optimal transport plans follow by a simple application of our general theory. The corresponding limit distribution is usually non-Gaussian which stands in strong contrast to recent finding for empirical entropy regularized optimal transport solutions.