Idempotent Addition Research Articles

Algebraic Solution of Tropical Best Approximation Problems

We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and output of an unknown function defined on an idempotent semifield, the problem is to find a best approximation of the function, by tropical Puiseux polynomial and rational functions. A new solution approach is proposed, which involves the reduction of the problem of polynomial approximation to the best approximate solution of a tropical linear vector equation with an unknown vector on one side (a one-sided equation). We derive a best approximate solution to the one-sided equation, and we evaluate the inherent approximation error in a direct analytical form. Furthermore, we reduce the rational approximation problem to the best approximate solution of an equation with unknown vectors on both sides (a two-sided equation). A best approximate solution to the two-sided equation is obtained in numerical form, by using an iterative alternating algorithm. To illustrate the new technique developed, we solve example approximation problems in terms of a real semifield, where addition is defined as maximum and multiplication as arithmetic addition (max-plus algebra), which corresponds to the best Chebyshev approximation by piecewise linear functions.

Semigroup models for biochemical reaction networks

The catalytic reaction system (CRS) formalism by Hordijk and Steel is a versatile method to model autocatalytic biochemical reaction networks. It is particularly suited, and has been widely used, to study self-sustainment and self-generation properties. Its distinguishing feature is the explicit assignment of a catalytic function to chemicals that are part of the system. In this work, it is shown that the subsequent and simultaneous catalytic functions give rise to an algebraic structure of a semigroup with the additional compatible operation of idempotent addition and a partial order. The aim of this article is to demonstrate that such semigroup models are a natural setup to describe and analyze self-sustaining CRS. The basic algebraic properties of the models are established and the notion of the function of any set of chemicals on the whole CRS is made precise. This leads to a natural discrete dynamical system on the power set of chemicals, which is obtained by iteratively considering the self-action on a set of chemicals by its own function. The fixed points of this dynamical system are proven to correspond to self-sustaining sets of chemicals, which are functionally closed. Finally, as the main application, a theorem on the maximal self-sustaining set and a structure theorem on the set of functionally closed self-sustaining sets of chemicals are proven.

Generalizations of the steffensen integral inequality for pseudo-integrals

Abstract In this paper, our aim is to prove certain kinds of Steffensen type integral inequalities for the pseudo-integral and the discrete pseudo-integral. The observations concern two cases of the real semiring with pseudo-operations with respect to pseudo-integrals: the first semiring, where pseudo-operations are defined via a monotone and continuous function g, the second semiring, when pseudo-operations are given by an idempotent addition and a generated pseudo-multiplication. Moreover, the discrete pseudo-integral is based on symmetric pseudo-addition and pseudo-multiplication, where the generator g is odd and increasing. In each case, several practical examples are presented to illustrate these results.

Algebraic Solution of Tropical Polynomial Optimization Problems

We consider constrained optimization problems defined in the tropical algebra setting on a linearly ordered, algebraically complete (radicable) idempotent semifield (a semiring with idempotent addition and invertible multiplication). The problems are to minimize the objective functions given by tropical analogues of multivariate Puiseux polynomials, subject to box constraints on the variables. A technique for variable elimination is presented that converts the original optimization problem to a new one in which one variable is removed and the box constraint for this variable is modified. The novel approach may be thought of as an extension of the Fourier–Motzkin elimination method for systems of linear inequalities in ordered fields to the issue of polynomial optimization in ordered tropical semifields. We use this technique to develop a procedure to solve the problem in a finite number of iterations. The procedure includes two phases: backward elimination and forward substitution of variables. We describe the main steps of the procedure, discuss its computational complexity and present numerical examples.

Algebraic Solution of a Problem of Optimal Project Scheduling in Project Management

A problem of the optimal scheduling is considered for a project that consists of a certain set of works to be performed under given constraints on the time of starting and finishing the works. The deviation of the starting time of the works that needs to be minimized is taken as the optimality criterion for scheduling. Such problems arise in project management when it is required, due to technological, organizational, economic, or other reasons, to ensure, wherever possible, that all works start simultaneously. The scheduling problem under consideration is formulated as a constrained minimax optimization problem and then solved using methods of tropical (idempotent) mathematics, which deals with the theory and applications of semirings with idempotent addition. First, a tropical optimization problem is investigated, which is defined in terms of a general idempotent semifield (an idempotent semiring with invertible multiplication), and a complete analytical solution of the problem is derived. The result obtained is then applied to find a direct solution of the scheduling problem in a compact vector form ready for the further analysis of the solutions and straightforward computations. As an illustration, a numerical example of solving a problem of optimal scheduling is given for a project that consists of four works.

Algebraic solution of a problem of optimal project scheduling in project management

A problem of optimal scheduling is considered for a project that consists of a certain set of works to be performed under given constraints on the times of start and finish of the works. As the optimality criterion for scheduling, the maximum deviation of the start time of works is taken to be minimized. Such problems arise in project management when it is required, according to technological, organizational, economic or other reasons, to provide, wherever possible, simultaneous start of all works. The scheduling problem under consideration is formulated as a constrained minimax optimization problem and then solved using methods of tropical (idempotent) mathematics which deals with the theory and applications of semirings with idempotent addition. First, a tropical optimization problem is investigated defined in terms of a general idempotent semifield (an idempotent semiring with invertible multiplication), and a complete analytical solution of the problem is derived. The result obtained is then applied to find a direct solution of the scheduling problem in a compact vector form ready for further analysis of solutions and straightforward computations. As an illustration, a numerical example of solving optimal scheduling problem is given for a project that consists of four works.

Anti-unification and the theory of semirings

It was recently shown that anti-unification over an equational theory consisting of only unit equations (more than one) is nullary. Such pure theories are artificial and are of little effect on practical aspects of anti-unification. In this work, we extend these nullarity results to the theory of semirings, a heavily studied theory with many practical applications. Furthermore, our argument holds over semirings with commutative multiplication and/or idempotent addition. We also cover a few open questions discussed in previous work.

Algebraic solution of minimax single-facility constrained location problems with Chebyshev and rectilinear distances

We consider location problems to find the optimal sites of placement of a new facility, which minimize the maximum weighted Chebyshev or rectilinear distance to existing facilities under constraints on a feasible location domain. We examine Chebyshev location problems in multidimensional space to represent and solve the problems in the framework of tropical (idempotent) algebra, which deals with the theory and applications of semirings and semifields with idempotent addition. The solution approach involves formulating the problem as a tropical optimization problem, introducing a parameter that represents the minimum value of the objective function in the problem, and reducing the problem to a system of parametrized inequalities. The necessary and sufficient conditions for the existence of a solution to the system serve to evaluate the minimum, whereas all corresponding solutions of the system present a complete solution of the optimization problem. With this approach we obtain direct, exact solutions represented in a compact closed form which is appropriate for further analysis and straightforward computations with polynomial time complexity. The solutions of the Chebyshev problems are then used to solve location problems with rectilinear distance in the two-dimensional plane. The obtained solutions extend previous results on the Chebyshev and rectilinear location problems without weights and with less general constraints.

Конечные циклические полукольца с полурешеточным сложением, заданным двухпорожденным идеалом натуральных чисел

В работе исследуются конечные циклические полукольца с~полурешеточным сложением, определенные как конечные циклические мультипликативные моноиды~$\langle S,\cdot \rangle$ с~введенной на них операцией сложения~$(+)$, так, что алгебраическая структура~$\langle S,+ \rangle$ является верхней полурешеткой и~выполняются законы дистрибутивности умножения относительно сложения.Описано строение конечных циклических полуколец с~полурешеточной аддитивной операцией, заданной двухпорожденным идеалом полукольца целых неотрицательных чисел.Результатом работы является теорема о строении циклических полуколец с~полурешеточной аддитивной операцией, заданной двухпорожденным идеалом полукольца целых неотрицательных чисел. Полученный результат, в частности, позволяет установить количество циклических полуколец, соответствующих каждому двухпорожденному идеалу полукольца целых неотрицательных чисел.В работе используется аппарат идеалов полукольца целых неотрицательных чисел.Получены некоторые свойства идеалов полукольца целых неотрицательных чисел, определяющих структуру конечных циклических полуколец.Работа дополняет исследования Е. М. Вечтомова и И. В. Орловой, где строение конечных циклических полуколец с~идемпотентным некоммутативным сложением описано через конечные циклические полуполя и~конечные циклические полукольца с~полурешеточным сложением.

Approximate factorization of positive matrices by using methods of tropical optimization

The problem of rank-one factorization of positive matrices with missing (unspecified) entries is considered where a matrix is approximated by a product of column and row vectors that are subject to box constraints. The problem is reduced to the constrained approximation of the matrix, using the Chebyshev metric in logarithmic scale, by a matrix of unit rank. Furthermore, the approximation problem is formulated in terms of tropical mathematics that deals with the theory and applications of algebraic systems with idempotent addition. By using methods of tropical optimization, direct analytical solutions to the problem are derived for the case of an arbitrary positive matrix and for the case when the matrix does not contain columns (rows) with all entries missing. The results obtained allow one to find the vectors of the factor decomposition by using expressions in a parametric form which is ready for further analysis and immediate calculation. In conclusion, an example of approximate rank-one factorization of a matrix with missing entries is provided.

On the Rank-One Approximation of Positive Matrices Using Tropical Optimization Methods

An approach to the problem of rank-one approximation of positive matrices in the Chebyshev metric in logarithmic scale is developed in this work, based on the application of tropical optimization methods. The theory and methods of tropical optimization constitute one of the areas of tropical mathematics that deals with semirings and semifields with idempotent addition and their applications. Tropical optimization methods allow finding a complete solution to many problems of practical importance explicitly in a closed form. In this paper, the approximation problem under consideration is reduced to a multidimensional tropical optimization problem, which has a known solution in the general case. A new solution to the problem in the case when the matrix has no zero columns or rows is proposed and represented in a simpler form. On the basis of this result, a new complete solution of the problem of rank-one approximation of positive matrices is developed. To illustrate the results obtained, an example of the solution of the approximation problem for an arbitrary two-dimensional positive matrix is given in an explicit form.

Solving a tropical optimization problem with application to optimal scheduling

A multidimensional optimization problem is formulated and solved in terms of tropical mathematics that is concerned with the theory and applications of semi-rings with idempotent addition. The problem, whose objective function is defined by a matrix, is proposed to be solved via idempotent algebra and tropical optimization tools. A strict lower bound is first derived for the objective function, used for solving the problem, to allow the evaluation of its minimum value. The objective function and its minimum value are then combined into an equation whose complete solution is obtained in the form of all eigenvectors of the matrix. A practical application of the problem is considered using the example of an explicit solution for the optimal scheduling of a project that consists of a set of activities defined by constraints on their start and end times. The optimality criterion for scheduling is defined to minimize the maximum, over all activities, of the working cycle time, which is described as the time interval between the start and the end of the activity. The analytical result extends and supplements the existing algorithmic numerical solutions to optimal scheduling problems. As an illustrative example, the solution of a problem to schedule a project consisting of three activities is presented to illustrate the result.

Using tropical optimization techniques in bi-criteria decision problems

We consider decision problems of rating alternatives based on their pairwise comparisons according to two criteria. Given pairwise comparison matrices for each criterion, the problem is to find the overall scores of the alternatives. We offer a solution that involves the minimax approximation of the comparison matrices by a common consistent matrix of unit rank in terms of the Chebyshev metric in logarithmic scale. The approximation problem reduces to a bi-objective optimization problem to minimize the approximation errors simultaneously for both comparison matrices. We formulate the problem in terms of tropical (idempotent) mathematics, which focuses on the theory and applications of algebraic systems with idempotent addition. To solve the optimization problem obtained, we apply methods and results of tropical optimization to derive a complete Pareto-optimal solution in a direct explicit form ready for further analysis and straightforward computation. We then exploit this result to solve the bi-criteria decision problem of interest. As illustrations, we present examples of the solution of two-dimensional optimization problems in general form, and of a decision problem with four alternatives in numerical form.

Complete algebraic solution of multidimensional optimization problems in tropical semifield

We consider multidimensional optimization problems that are formulated in the framework of tropical mathematics to minimize functions defined on vectors over a tropical semifield (a semiring with idempotent addition and invertible multiplication). The functions, given by a matrix and calculated through multiplicative conjugate transposition, are nonlinear in the tropical mathematics sense. We start with known results on the solution of the problems with irreducible matrices. To solve the problems in the case of arbitrary (reducible) matrices, we first derive the minimum value of the objective function, and find a set of solutions. We show that all solutions of the problem satisfy a system of vector inequalities, and then use these inequalities to establish characteristic properties of the solution set. Furthermore, all solutions of the problem are represented as a family of subsets, each defined by a matrix that is obtained by using a matrix sparsification technique. We describe a backtracking procedure that allows one to reduce the brute-force generation of sparsified matrices by skipping those, which cannot provide solutions, and thus offers an economical way to obtain all subsets in the family. Finally, the characteristic properties of the solution set are used to provide complete solutions in a closed form. We illustrate the results obtained with simple numerical examples.

Lattice-ordered abelian groups finitely generated as semirings

A lattice-ordered group (an $\ell$-group) $G(\oplus, \vee, \wedge)$ can be naturally viewed as a semiring $G(\vee,\oplus)$. We give a full classification of (abelian) $\ell$-groups which are finitely generated as semirings, by first showing that each such $\ell$-group has an order-unit so that we can use the results of Busaniche, Cabrer and Mundici [8]. Then we carefully analyze their construction in our setting to obtain the classification in terms of certain $\ell$-groups associated to rooted trees (Theorem 4.1). This classification result has a number of important applications: for example it implies a classification of finitely generated ideal-simple (commutative) semirings $S(+, \cdot)$ with idempotent addition and provides important information concerning the structure of general finitely generated ideal-simple (commutative) semirings, useful in obtaining further progress towards Conjecture 1.1 discussed in [2], [15].

Using tropical optimization to solve minimax location problems with a rectilinear metric on the line

Methods of tropical (idempotent) mathematics are applied to the solution of minimax location problems under constraints on the feasible location region. A tropical optimization problem is first considered, formulated in terms of a general semifield with idempotent addition. To solve the optimization problem, a parameter is introduced to represent the minimum value of the objective function, and then the problem is reduced to a parametrized system of inequalities. The parameter is evaluated using existence conditions for solutions of the system, whereas the solutions of the system for the obtained value of the parameter are taken as the solutions of the initial optimization problem. Then, a minimax location problem is formulated to locate a single facility on a line segment in the plane with a rectilinear metric. When no constraints are imposed, this problem, which is also known as the Rawls problem or the messenger boy problem, has known geometric and algebraic solutions. For the location problems, where the location region is restricted to a line segment, a new solution is obtained, based on the representation of the problems in the form of the tropical optimization problem considered above. Explicit solutions of the problems for various positions of the line are given both in terms of tropical mathematics and in the standard form.

A Rigourous Mathematical Framework for Computing a Sustainability Ratio: the Emergy

The computational problem of emergy within a general system of interconnected processes at steady state is a subject of interest in literature today. When there is no co-product the proposed method coincides with the Track Summing Method of Tennenbaum which was developed precisely for interconnected networks with feedbacks and splits of emergy. As the underlying algebraic structure of the Tennenbaum's method is the linear algebra, it is not well-suited to account for the co-product problem which induces the idempotent operator max. Thus, authors have chosen another underlying algebraic structure which is the idempotent semiring structure (i.e. a semiring equipped with an idempotent addition). This method is divided into two parts. The first part is the emergy flow enumeration, where paths from an emergy source to the input of a given process are enumerated avoiding double counting of emergy assignation. This part is a path-finding problem which is a slight modification of gerbier of null square approach to find elementary/simple paths in a graph. The second part evaluates the emergy flowing between two components of the system. It is a quantitative part in which the problem of avoiding double counting split and co-product flows are dealt with by introducing a way to mark splits and co-products flows. The method is partially parallelizable. However, the method enumerates paths on a graph thus, in worst cases, its complexity is not polynomial. This paper provides a rigorous framework based on an axiomatic basis to conduct the emergy evaluation of an emergy graph. © 2012 ISEIS All rights reserved.

Duality and interval analysis over idempotent semirings

In this paper semirings with an idempotent addition are considered. These algebraic structures are endowed with a partial order. This allows to consider residuated maps to solve systems of inequalities A⊗X⪯B (see [3]). The purpose of this paper is to consider a dual product, denoted ⊙, and the dual residuation of matrices, in order to solve the following inequality A⊗X⪯X⪯B⊙X. Sufficient conditions ensuring the existence of a non-linear projector in the solution set are proposed. The results are extended to semirings of intervals such as they were introduced in [25].

Generalization of the Stolarsky type inequality for pseudo-integrals

In this paper, we prove a Stolarsky type inequality for two classes of pseudo-integrals. One of them deals with pseudo-integrals where pseudo-operations are defined via a monotone and continuous generator function. The other one concerns the pseudo-integrals based on a semiring with an idempotent addition and a generated pseudo-multiplication.

A class of inherently nonfinitely based semirings

We give a sufficient condition which ensures that a semiring with an idempotent addition is inherently nonfinitely based. This enables us to provide a number of small and natural examples of nonfinitely based semirings, including semirings of binary relations on a finite set.