Connectionist networks for pivot selection in linear programming

The study deals with the multi-choice mathematical programming problem, where the right hand side of the constraints is multi-choice in nature. However, the problem of multi-choice linear programming cannot be solved directly by standard linear or nonlinear programming techniques. The aim of this paper is to transform such problems to a standard mathematical linear programming problem. For each constraint, exactly one parameter value is selected out of a multiple number of parameter values. This process of selection can be established in different ways. In this paper, we present a new simple technique enabling us to handle such problem as a mixed integer linear programming problem and consequently solve them by using standard linear programming software. Our main aim depends on inserting a specific number of binary variables and using them to construct a linear combination which gives just one parameter among the multiple choice values for each choice of the values of the binary variables. A numerical example is presented to illustrate our analysis.

We formally derive the standard deterministic linear program (LP) for bid-price control by making an affine functional approximation to the optimal dynamic programming value function. This affine functional approximation gives rise to a new LP that yields tighter bounds than the standard LP. Whereas the standard LP computes static bid prices, our LP computes a time trajectory of bid prices. We show that there exist dynamic bid prices, optimal for the LP, that are individually monotone with respect to time. We provide a column generation procedure for solving the LP within a desired optimality tolerance, and present numerical results on computational and economic performance.

We study the ATSP (Asymmetric Traveling Salesman Problem), and our focus is on negative results in the framework of the Sherali-Adams (SA) Lift and Project method. Our main result pertains to the standard LP (linear programming) relaxation of ATSP, due to Dantzig, Fulkerson, and Johnson. For any fixed integer t≥0 and small e, 0<e≪1, there exists a digraph G on ν=ν(t,e)=O(t/e) vertices such that the integrality ratio for level t of the SA system starting with the standard LP on G is ≥ 1+ 1-&epsilon/2t+3 ≈ 4/3, 6/5, 8/7, …. Thus, in terms of the input size, the result holds for any t=0,1,…,Θ(ν) levels. Our key contribution is to identify a structural property of digraphs that allows us to construct fractional feasible solutions for any level t of the SA system starting from the standard LP. Our hard instances are simple and satisfy the structural property. There is a further relaxation of the standard LP called the balanced LP, and our methods simplify considerably when the starting LP for the SA system is the balanced LP; in particular, the relevant structural property (of digraphs) simplifies such that it is satisfied by the digraphs given by the well-known construction of Charikar, Goemans and Karloff (CGK). Consequently, the CGK digraphs serve as hard instances, and we obtain an integrality ratio of $1 +\frac{1-\epsilon}{t+1}$ for any level t of the SA system, where 0<e≪1 and the number of vertices is ν(t,e)=O((t/e)(t/e)). Also, our results for the standard LP extend to the path ATSP (find a min cost Hamiltonian dipath from a given source vertex to a given sink vertex).

Individual or cooperation as the producer who produces more than one product always has a problem to determine the amount of each product, as well as the smallest packaging used to generate the maximum revenue but still meet the inventory of existing facilities. If the inventory of facilities and functions form a linear function of income then the problem is called the program a maximu m linear case, where all supplies are limited means of the standard linear program is called maximal. Although the system has available a linear program to solve the problem, the system is only owned by a limited circle, this system also does not provide the smallest packaging option. In general linear programming problem is solved by the simplex method with any number of real output in decimal or fractional format mix. Settlement is not operational because the manufacturers do not always able to make the smallest product packaging in accordance with the unit on completion, to become operational, the results of this settlement may be rounded according to the smallest package selected. The problem of limited manufactures which have a system of linear program application can be overcome by providing a system similar to a web-based applications. This study compiled a web-based application system to solve a standard linear program that generates the maximum revenue that maximum use PHP software, data entered through the form and then taken by the POST method, then set up a matrix, carried out by iterating between the rows in the matrix operations that met the criteria optimal. By ch oosing the smallest packaging which is used 0.25 units or 0.50 units, the system check the number of products that have the smallest rounding according to the package selected, then the system prepare and present the inventory and revenue report. Keywords: Simplex metho; The smallest packaging; Web-based

Solving (mixed) integer (linear) programs, (M)I(L)Ps for short, is a fundamental optimisation task with a wide range of applications in artificial intelligence and computer science in general. While hard in general, recent years have brought about vast progress for solving structurally restricted, (non-mixed) ILPs: n-fold, tree-fold, 2-stage stochastic and multi-stage stochastic programs admit efficient algorithms, and all of these special cases are subsumed by the class of ILPs of small treedepth. In this paper, we extend this line of work to the mixed case, by showing an algorithm solving MILP in time f(a,d)poly(n), where a is the largest coefficient of the constraint matrix, d is its treedepth, and n is the number of variables. This is enabled by proving bounds on the denominators (fractionality) of the vertices of bounded-treedepth (non-integer) linear programs. We do so by carefully analysing the inverses of invertible sub-matrices of the constraint matrix. This allows us to afford scaling up the mixed program to the integer grid, and applying the known methods for integer programs. We then trace the limiting boundary of our "bounded fractionality" approach both in terms of going beyond MILP (by allowing non-linear objectives) as well as its usefulness for generalising other important known tractable classes of ILP. On the positive side, we show that our result can be generalised from MILP to MIP with piece-wise linear separable convex objectives with integer breakpoints. On the negative side, we show that going even slightly beyond such objectives or considering other natural related tractable classes of ILP leads to unbounded fractionality. Finally, we show that restricting the structure of only the integral variables in the constraint matrix does not yield tractable special cases.

Probabilistic programming is used in some optimization problems where some or all parameters are considered as random variables, in order to deal with uncertainty, which is an inherent feature of the system. The situation of multiple parameters may exist in a decision making problem in our real life. The multi-choice programming can not only avoid the underestimation of parameters, but also can decide the appropriate parameter from multiple parameters. This paper deals with a probabilistic linear programming problem, where the right hand side parameters of probabilistic constraints are multi-choice in nature and rest of the parameters are independent random variables. In this paper the probabilistic programming problem is converted to an equivalent deterministic mathematical programming model. The resulting model is then solved by standard linear or non-linear programming techniques. A numerical example is presented to illustrate the methodology.

This paper studies the problem of tangible assets acquisition within the company by proposing a new hybrid model that uses linear programming and fuzzy numbers. Regarding linear programming, two methods were implemented in the model, namely: the graphical method and the primal simplex algorithm. This hybrid model is proposed for solving investment decision problems, based on decision variables, objective function coefficients, and a matrix of constraints, all of them presented in the form of triangular fuzzy numbers. Solving the primal simplex algorithm using fuzzy numbers and coefficients, allowed the results of the linear programming problem to also be in the form of fuzzy variables. The fuzzy variables compared to the crisp variables allow the determination of optimal intervals for which the objective function has values depending on the fuzzy variables. The major advantage of this model is that the results are presented as value ranges that intervene in the decision-making process. Thus, the company’s decision makers can select any of the result values as they satisfy two basic requirements namely: minimizing/maximizing the objective function and satisfying the basic requirements regarding the constraints resulting from the company’s activity. The paper is accompanied by a practical example.

The Nemhauser–Trotter theorem states that the standard linear programming (LP) formulation for the stable set problem has a remarkable property, also known as (weak) persistency: for every optimal LP solution that assigns integer values to some variables, there exists an optimal integer solution in which these variables retain the same values. While the standard LP is defined by only non-negativity and edge constraints, a variety of other LP formulations have been studied and one may wonder whether any of them has this property as well. We show that any other formulation that satisfies mild conditions cannot have the persistency property on all graphs, unless it is always equal to the stable set polytope.

In this article, we present a new approach to the solution of optimization problems with discrete control variables modeled by binary integer programs (BIP). The solution of BIP is computationally demanding when the number of the BIP variables increase. Two instances that increase the number BIP variables in practical applications are the reduction of the discretization sampling time and the increase of the optimization time period. The proposed approach transforms a single BIP optimization into a linear program (LP) and N feasibility BIP's, with less number of variables. The reduction of the number of variables increases the algorithm speed in providing a solution. The approach permits to solve optimization problems with longer time intervals and with a higher number of control variables, while being computationally tractable. A case study on the electricity cost minimization in a pumping station shows the applicability of the method.

This article addresses a special class of non-linear programming problems, viz., linear fractional programming problems having multiple objectives. In solving the real-life linear fractional optimization problems, the ambiguity and hesitation in the decision are inherent and ever-present, therefore, it is perfectly viable to formulate and solve these optimization models using the intuitionistic fuzzy environment. The purpose of this study is to propose a simple and computationally efficient approach to obtain the solution of multiple objective linear fractional programming problems having all the decision variables and parameters expressed in terms of triangular intuitionistic fuzzy numbers. The proposed solution algorithm is primarily based on the goal programming approach, fuzzy-based linearization technique, and a membership function strategy. The original linear fractional programming problem is first converted to its equivalent deterministic/crisp multi-objective linear fractional optimization problem using the weighted goal programming methodology along with the linear membership technique to resolve the intuitionistic fuzzy constraints into the crisp one. Finally, the variable transformation technique for the under- and over-deviational variables of the goal programming model is employed to linearize all the fractions involved in the problem so as to convert the original problem to an equivalent linear optimization problem. Further, this linear programming problem can be solved using any available commercial packages. Moreover, a numerical illustration is provided to demonstrate the steps of the proposed technique followed by the analysis and solution of an E-education set-up problem. The discussion and comparisons of the practical case establish the relevancy and usefulness of the proposed model.

Population-based approaches have given us new search strategies and ideas in order to solve optimization problems. Usually, these methods are based on the performance carried out by a finite number of agents, which by the interaction between them they evolve and work all over the search space. Also, it is well-known that the correct employment of parameter values in this kind of method can positively impact their performance and behavior. In this context, the present work focuses on the design of a hybrid architecture which smartly balances the population size on run-time. In order to smartly balance and control the population size, a modular approach, named Linear Modular Population Balancer (LMPB), is proposed. The main ideas behind the designed architecture include the solving strategy behind a population-based metaheuristic, the influence of learning components based on multiple statistical modeling methods which transform the dynamic data generated into knowledge, and the possibilities to tackle both discrete and continuous optimization problems. In this regard, three modules are proposed for LMPB, which concern tasks such as the management of the population-based algorithm, parameter setting, probabilities, learning methods, and selection mechanism for the population size to employ. In order to test the viability and effectiveness of our proposed approach, we solve a set of well-known benchmark functions and the multidimensional knapsack problem (MKP). Additionally, we illustrate promising solving results, compare them against state-of-the-art methods which have proved to be good options for solving optimization problems, and give solid arguments for future work in the necessity to keep evolving this type of proposed architecture.

We investigate the problem of optimally placing limited-range wavelength converters at a subset of nodes in wavelength-routed wavelength-division-multiplexing (WDM) networks. We consider two different aspects of the converter-placement problem: 1) the placement minimizing the network-wide blocking probability; and 2) the placement minimizing the number of wavelength-convertible nodes to meet the performance constraints. We present a binary (0-1) linear program (BLP) formulation in which the end-to-end blocking probability is expressed as a linear function of converter locations so that standard linear program (LP) optimization packages can be employed to obtain the optimal solution of the problems. We also present a new analytical model for estimating the end-to-end blocking probabilities. Experiments have been conducted over four network topologies, including 19-node European optical network (EON), 24-node USA backbone network (UBN), 32-node HYPERCUBE, and 36-node MESH-TORUS. We demonstrate that the optimal solutions of the converter-placement problems can be obtained within a reasonable computation time.

This paper establishes a framework for solving some optimization problems with linear constraints using simplex-type methods. The problems include those found in linear programs, linear fractional programs, and generalized linear fractional programs. In this study, these problems refer to a standard form of minimizing a single parameter subject to parameterized linear equations. Based on the analysis of parameterized basis-based solutions, a unified simplex-type approach is proposed. The adaptability of the parameterized model and that of the solution procedure are discussed. In particular, the proposed algorithm can prevent cycling when compared with the conventional simplex method used for solving linear programs.

In many recent works, the potential of Exploratory Landscape Analysis (ELA) features to numerically characterize single-objective continuous optimization problems has been demonstrated. These numerical features provide the input for all kinds of machine learning tasks in the domain of continuous optimization problems, ranging, for example, from High-level Property Prediction to Automated Algorithm Selection and Automated Algorithm Configuration. Without ELA features, analyzing and understanding the characteristics of single-objective continuous optimization problems is-to the best of our knowledge-very limited. Yet, despite their usefulness, as demonstrated in several past works, ELA features suffer from several drawbacks. These include, in particular, (1) a strong correlation between multiple features, as well as (2) its very limited applicability to multiobjective continuous optimization problems. As a remedy, recent works proposed deep learning-based approaches as alternatives to ELA. In these works, among others, point-cloud transformers were used to characterize an optimization problem's fitness landscape. However, these approaches require a large amount of labeled training data. Within this work, we propose a hybrid approach, Deep-ELA, which combines (the benefits of) deep learning and ELA features. We pre-trained four transformers on millions of randomly generated optimization problems to learn deep representations of the landscapes of continuous single- and multiobjective optimization problems. Our proposed framework can either be used out of the box for analyzing single- and multiobjective continuous optimization problems, or subsequently fine-tuned to various tasks focusing on algorithm behavior and problem understanding.

This book is on modern continuous nonlinear optimization. Continuous nonlinear optimization problems have a simple mathematical model and always refer to a real physical system, the running of which we want to optimize. Firstly, a nonlinear optimization problem contains an objective function which measures the performances or requirements of the system. Often, this function represents a profit, a time interval, a level, a sort of energy or combination of different quantities which have a physical significance for the modeler. The objective function depends on some characteristics of the system, called variables or unknowns. The purpose of any optimization problem is to find the values of these variables that minimize (or maximize) the objective function subject to some constraints which the variables must satisfy. Constraints of an optimization problem may have different algebraic expressions. There are static and dynamic constraints called functional constraints. The difference between these types of constraints comes from the structure of their Jacobian. The dynamic constraints always involve a Jacobian with a block-diagonal structure. Besides, these functional constraints are nonlinear functions which may be equalities or inequalities, or even range constraints. Another very important type of constraints is the simple bounds on variables. The real applications of optimization include simple bounds on variables which express the constructive engineering conditions and limits of the system to be optimized. The constraints of an optimization problem define the so-called feasible domain of the problem. Both the objective function and the constraints may depend on some parameters with known values which characterize the system under optimization. The process of identifying the variables, parameters, simple bounds on variables, the objective functions, and constraints is known as modeling, one of the finest intellectual activities. In this book we assume that the variables can take real values and the objective function and the constraints are smooth enough (at least twice continuously differentiable) with known first-order derivatives. When the number of variables and the number of constraints is large, then the optimization problem is quite challenging.