Continuous Linear Programming Problems Research Articles

Efficient and robust optimal design for quantile regression based on linear programming

When informing decisions with experimental data, it is often necessary to quantify the distribution tails of uncertain system responses using limited data. To maximize the information content of the data, one is naturally led to use experimental design. However, common design techniques minimize global statistics such as the average estimation or prediction variance. Novel methods for optimal experimental design that target distribution tails are developed. To achieve this, pre-asymptotic estimates of the data uncertainty are produced via an upper bound on a prescribed quantile, computed using quantile regression. Two optimal design problems are formulated: (i) Minimize the variance of the upper bound; and (ii) Minimize the Conditional Value-at-Risk of the upper bound. Additionally, each design problem is augmented with an added cardinality constraint to bound the number of experiments. These optimal design problems are reduced to continuous and mixed-integer linear programming problems. Consequently, the proposed methods are extremely efficient, even when applied to large datasets. The application of the proposed design formulation is demonstrated through a sensor placement problem in direct field acoustic testing.

Optimal operation planning of distribution systems under uncertainty in load modeling by using decentralized energy resources

The new elements connected in electrical distribution systems increase the complexity of grids as well as planning and operating. The benefits of classical techniques, such as Conservation Voltage Reduction (CVR), combined with a coordinated operation of distribution elements, can contribute to increase efficiency and to reduce energy consumption of the distribution systems.This paper proposes an optimization model for a day-ahead operation of an unbalanced distribution system with photovoltaic distributed generation (DG) penetration. The CVR technique will be considered in deterministic, stochastic and robust approaches, considering the uncertainty of its parameters in the load modeling. The proposed optimization model aims to minimize energy consumption at the substation considering the operation of traditional elements, such as On Load Tap Changers (OLTC) at substation and capacitor banks (CB), but also modern elements, such as intelligent photovoltaic inverters.In the proposed approach, the mathematical problem, initially modelled as a mixed-integer non-linear programming problem, was converted into a continuous linear programming problem. The results were obtained using a modified IEEE123-bus test system indicating good results for utilities to save energy and reduce power demand in three optimization approaches: deterministic, stochastic and robust, and the last one proved to be better for being conservative for voltage violations and lower computational burden.

FACTORING MANAGEMENT ON THE BASIS OF FORMATION OF OPTIMIZING FACTORING PORTFOLIO OF TRADE ENTERPRISES

Abstract. The article is devoted to the study of the effectiveness of factoring management as an innovative product based on the formation of the optimization factoring portfolio of trade enterprises, which will serve as a basis for management decisions on the implementation of this service in business. As a result of the research a number of conclusions and methodical recommendations of practical direction are formulated. The formation of the optimization factoring portfolio is based on the principles of the CAPM portfolio investment model proposed by W. Sharp. The conditions for the formation of portfolio investments, which are reflected in the price model of the capital market, were adapted to the formation of the factoring portfolio. The key conditions of the factoring agreement are: limits on credit risk and a given level of income, that is the company must choose a portfolio with the maximum profitability, which is achieved by minimizing the cost of this service; seek to increase the funding limit for debtors; to remove from the portfolio of debtors of the enterprise with a high risk of non-repayment of debt. We propose to determine the debtors of the enterprise that will form the optimization factoring portfolio through the variables x1 – xn. When x = 1, the debtor is added to the factoring portfolio, and if x = 0, then it is necessary to exclude the debtor from the portfolio. In the language of factoring regression agreements, you can use a continuous linear programming problem in which the variable x will be in the range from 0 to 1, that is funding limits for debtors can be broken. The target function of the factoring portfolio of commercial enterprises is proposed both for «invoice-discounting» (only receivables financing) and for «full factoring», that is full-fledged factoring (the use of additional related factoring services). The profitability of the enterprise from factoring operations depends on the number of debtors in the factoring portfolio. The greater the number of debtors in the portfolio, the higher the profitability of the company from factoring. However, the company needs to consider debtors with high credit risk in order to minimize the non-repayment of receivables. Therefore, debtors with the highest credit risk of factoring should be excluded from the factoring portfolio. To assess the economic efficiency of factoring management of trade enterprises on the basis of the formed optimization factoring portfolios, we propose to use the Cronbach & Gleser method. The proposed proposals for the management of factoring of commercial enterprises, which is based on the formation of factoring portfolios, will regulate debt relations between enterprises with the involvement of third parties, that is, to prevent the occurrence of negative phenomena in the debt relations between commercial enterprises in the future, as well as will help maximize profits in the studied business entities. Further research will focus on adapting the factoring portfolio of commercial enterprises to changes in the value of financial resources under the influence of transformational changes in the economy. Keywords: factoring portfolio, trading enterprises, transformational changes, factoring management, debt relations between enterprises, profit maximization. JEL Classification С44, С65, С61, F65 Formulas: 11; fig.: 0; tabl.: 0; bibl.: 20.

A two-level vertex-searching global algorithm framework for bilevel linear fractional programming problems

ABSTRACT We propose a new two-level vertex-searching algorithm framework that finds a global optimal solution to the continuous bilevel linear fractional programming problem over a compact polyhedron, in which both the upper and the lower objectives are linear fractional. Our solution method adopts the vertex-searching approach on the polyhedron, and the search space is determined by the set of candidates of optimal base to the lower level problem. In order to search base, a modified enumerative scheme, that is a new upper bound filter scheme inserted into the classical enumerative scheme, is proposed. The main solution procedure is designed on solving a sequence of upper and lower level mathematical programs; instead of a single-level problem reformulation approach, which is popularly and widely used in literature. An extension on general upper level objective functions such as quasiconvex/quasiconcave for the proposed vertex-searching approach is discussed. Numerical experiments show that our algorithm leads us to a global optimum. We conclude that our proposed algorithm framework has the simplest solution procedure and has potential efficiency advantages, which may reduce the complexity of enumerative schemes for medium or large-scale problems, while comparing with existing global algorithms such as the Kth-best algorithm (a two-level vertex-searching algorithm) and the single-level duality-based reformulation algorithm (a single-level vertex-searching algorithm).

A multi-user multi-operator computing pricing method for Internet of things based on bi-level optimization

The Internet of mobile things is a promising paradigm that generates, stores, and processes amount of real-time data to render rich services for mobile users. Along with the increase of mobile devices in the field of Internet of things, more and more intelligent applications, such as face recognition and virtual reality, have emerged. These applications typically consume large amounts of computing and energy resources. However, due to the physical size limitations of Internet of things terminals, their computing capacity and power are limited, where users’ needs for application processing delay and power consumption cannot be met. Therefore, the concept of edge cloud computing has been proposed, which enhances the computing capacity of Internet of things terminals by offloading user tasks to edge servers for computation. When there are multiple operators, it is important to understand how users choose an operator to perform computation and how operators can reasonably price the computing capacity to meet their own interests. Therefore, we study the computation pricing and user decision-making problems of Internet of things under multi-user and multi-operator scenarios. The problem is divided into three phases and modeled as a two-level optimization problem. While an operator’s goal is to minimize the loss of his interests, the user’s goal is to minimize the computation cost (energy consumption and price). First, since the lower-level user decision-making problem is an integer linear programming problem, we transform it into an equivalent continuous linear programming problem by relaxation. Second, we transform the bi-level optimization problem into an equivalent single-level optimization problem by substituting the lower problem’s Karush–Kuhn–Tucker conditions into an upper problem. Finally, we use a spatial branch and bound algorithm to solve the problem. Experimental results show that the proposed algorithm can effectively maintain the benefits of both operators and users in the field of Internet of things.

An effective method for optimization of continuous and discrete parameters of heat and power plants

Modern thermal power plants are complex technological systems. Therefore, making informed decisions when studying them requires the use of mathematical modeling and nonlinear optimization methods for plant parameter. The most complex task is to solve a mixed optimization problem wherein a part of optimization parameters vary continuously, and the other can take only discrete (integer) values. An effective method is developed to solve a thermal power plant optimization problem with continuous and discrete parameters. The method suggests an iterative procedure for solving continuous nonlinear programming problems and discrete-continuous linear programming problems. For each iteration, we add new constraints obtained by linearizing nonlinear inequality constraints and the objective function of the initial problem to the system of inequality constraints of linear problem. The effectiveness of the proposed method is exemplified by the optimization of a combined cycle power plant with a mixture of working media (the STIG scheme). Design characteristics of heating surface of a waste heat recovery boiler represent the discrete parameters to be optimized. The research demonstrates a considerable reduction in computational effort compared to the branch and bound method.

Why Unified Statistics Theory by MCMC Towards Linear and Nonlinear Programming Problems?

Unified statistics theory by MCMC is considered. A new proposed algorithm is presented to obtain surely empirical analysis conclusions in order to turn to surely theoretical analysis results about the behavior of any general linear or nonlinear programming problem in order to introduce a complete framework and to solve any too large dimensional deterministic and probabilistic (the grouping data, both continuous and discrete) linear or nonlinear programming problems by the proposed algorithm that has two obvious criteria towards the second resounding success of unified statistics theory by MCMC.

Conflict graphs in implicit enumeration

A zero-one linear program is a global discrete optimization problem. Namely, it is a linear programming problem with zero-one variables. The real relaxation of a zero-one program is again a continuous linear programming problem. Simply, the zero-one variables are replaced by continuous variables varying between zero and one independently of each other. If the real relaxation, as a continuous problem, solved with the simplex method happens to have a zero-one optimal solution, then this particular solution is also an optimal solution of the original zero-one programming problem. Suppose that x1,…,xn are all the variables of a given zero-one linear programming problem. For convenience we introduce further variables y1,…,yn defined by y1=1−x1,…, yn=1−xn respectively. For a unified notation the variables x1,…,xn and y1,…, yn together will be denoted by z1,…,z2n. These variables will play the roles the nodes of the conflict graph Г associated with the given zero-one linear programming problem. Fixing the variables zi and zj both to be equal to one simultaneously reduces the number of variables in the problem. If the new smaller optimization problem does not have any feasible solution, then the nodes zi and zj will be connected by an edge in the graph Γ. The graph Γ simply records that there is a conflict between the assignments zi =1 and zj =1 which explains the name conflict graph. In other words the non-directed edge between the nodes zi and zj codes the fact that the inequality zi+zj≤1holds. The conflict graph was designed to generate additional constraints the so-called valid inequalities or cuts in order to expedite the solution of the zero-one linear program via its real relaxation. This particular solution strategy is aptly named the branch and cut method. It will be shown that the conflict graph besides its intended use can also be applied in another solution technique the method of implicit enumeration. We will illustrate by examples that the conflict graph provides us with rules to prune the search tree that are not offered by the commonly applied pruning rules. In addition the computations involved can easily be organized into a highly parallel computational scheme.

A recurrence method for a special class of continuous time linear programming problems

This article studies a numerical solution method for a special class of continuous time linear programming problems denoted by (SP). We will present an efficient method for finding numerical solutions of (SP). The presented method is a discrete approximation algorithm, however, the main work of computing a numerical solution in our method is only to solve finite linear programming problems by using recurrence relations. By our constructive manner, we provide a computational procedure which would yield an error bound introduced by the numerical approximation. We also demonstrate that the searched approximate solutions weakly converge to an optimal solution. Some numerical examples are given to illustrate the provided procedure.

A RECURRENCE METHOD FOR SIMPLE CONTINUOUS LINEAR PROGRAMMING PROBLEMS

In this paper, we discuss a special class of continuous linear programming problems which can be called simple continuous linear programming problems (SP). A practical and efficient method for finding an approximate optimal value and optimal solution of (SP) is presented. The main work of computing an approximate optimal value in the provided method is only to solve finite linear programming problems by using recurrence relations. Furthermore, a simple algorithm can be employed not only to easily solve the (SP) problem but also to provide an error bound of optimal value as well. Some numerical examples are given to implement the algorithm.

A RECURRENCE METHOD FOR SIMPLE CONTINUOUS LINEAR PROGRAMMING PROBLEMS

ABSTRACT In this paper, we discuss a special class of continuous linear programming problems which can be called simple continuous linear programming problems (SP). A practical and efficient method for finding an approximate optimal value and optimal solution of (SP) is presented. The main work of computing an approximate optimal value in the provided method is only to solve finite linear programming problems by using recurrence relations. Furthermore, a simple algorithm can be employed not only to easily solve the (SP) problem but also to provide an error bound of optimal value as well. Some numerical examples are given to implement the algorithm.

A simplex based algorithm to solve separated continuous linear programs

We consider the separated continuous linear programming problem with linear data. We characterize the form of its optimal solution, and present an algorithm which solves it in a finite number of steps, using an analog of the simplex method, in the space of bounded measurable functions.

On the adjustment problem for linear programs

We propose a generalization of the inverse problem which we will call the adjustment problem. For an optimization problem with linear objective function and its restriction defined by a given subset of feasible solutions, the adjustment problem consists in finding the least costly perturbations of the original objective function coefficients, which guarantee that an optimal solution of the perturbed problem is also feasible for the considered restriction. We describe a method of solving the adjustment problem for continuous linear programming problems when variables in the restriction are required to be binary.

New Branch-and-Cut Algorithm for Bilevel Linear Programming

Linear mixed 0---1 integer programming problems may be reformulated as equivalent continuous bilevel linear programming (BLP) problems. We exploit these equivalences to transpose the concept of mixed 0---1 Gomory cuts to BLP. The first phase of our new algorithm generates Gomory-like cuts. The second phase consists of a branch-and-bound procedure to ensure finite termination with a global optimal solution. Different features of the algorithm, in particular, the cut selection and branching criteria are studied in details. We propose also a set of algorithmic tests and procedures to improve the method. Finally, we illustrate the performance through numerical experiments. Our algorithm outperforms pure branch-and-bound when tested on a series of randomly generated problems.

Efficient Algorithms for Separated Continuous Linear Programs: The Multicommodity Flow Problem with Holding Costs and Extensions

We give an approximation scheme for separated continuous linear programming problems. Such problems arise as fluid relaxations of multiclass queueing networks and are used to find approximate solutions to complex job shop scheduling problems. In a network with linear flow costs and linear, per-unit-time holding costs, our algorithm finds a drainage of the network that, for given constants ε > 0 and δ > 0, has total cost (1 + ε)OPT + δ, where OPT is the cost of the minimum cost drainage. The complexity of our algorithm is polynomial in the size of the input network, 1/ε, and log(1/δ). The fluid relaxation is a continuous problem. While the problem is known to have a piecewise constant solution, it is not known to have a polynomially sized solution. We introduce a natural discretization of polynomial size and prove that this discretization produces a solution with low cost. This is the first polynomial time algorithm with a provable approximation guarantee for fluid relaxations.

Generic Uniqueness of the Solutions to a Continuous Linear Programming Problem

Consider two continuous functions f,g mapping the interval [0,S] of the real line into R. Let f also be strictly increasing. We are interested in the set of probability distributions on the interval [0,S] that maximize the expectation of f subject to the constraint that the expectation of g be no greater than a constant. We provide a sufficient condition on the pair (f,g) for the solution to this linear programming problem to be unique and show that this sufficient condition is satisfied generically.

A penalty function heuristic for the resource constrained shortest path problem

The resource constrained shortest path problem (RCSP) consists of finding the shortest path between two nodes of an assigned network, with the constraint that traversing an arc of the network implies the consumption of certain limited resources. In this paper we propose a new heuristic for the solution of the RCSP problem in medium and large scale networks. It is based on the extension to the discrete case of the penalty function heuristic approach for the fast ε-approximate solution of difficult large-scale continuous linear programming problems. Computational experience on test instances has shown that the proposed penalty function heuristic (PFH) is very effective in the solution of medium and large scale RCSP instances. For all the tests reported it provides very good upper bounds (in many cases the optimal solution) in less than 26 iterations, where each iteration requires only the computation of a shortest path.

Uncertainty modelling and robust output feedback control of nonlinear discrete systems: a mathematical programming approach

We present a mathematical programming approach to robust control of nonlinear systems with uncertain, possibly time-varying, parameters. The uncertain system is given by different local affine parameter-dependent models in different parts of the state space. It is shown how this representation can be obtained from a nonlinear uncertain system by solving a set of continuous linear semi-infinite programming problems, and how each of these problems can be solved as a (finite) series of ordinary linear programs. Additionally, the system representation includes control and state constraints. The controller design method is derived from Lyapunov stability arguments and utilizes an affine parameter-dependent quadratic Lyapunov function. The controller has a piecewise affine output feedback structure, and the design amounts to finding a feasible solution to a set of linear matrix inequalities combined with one spectral radius constraint on the product of two positive definite matrices. A local solution approach to this non-convex feasibility problem is proposed. Complexity of the design method and some special cases such as state feedback are discussed. Finally, an application of the results is given by proposing an on-line computationally feasible algorithm for constrained nonlinear state-feedback model predictive control with robust stability. Copyright © 2000 John Wiley & Sons, Ltd.

Optimal Control of Semilinear Parabolic Equations with State-Constraints of Bottleneck Type

We consider optimal distributed and boundary control problems for semilinear parabolic equations, where pointwise constraints on the control and pointwise mixed control-state constraints of bottleneck type are given. Our main result states the existence of regular Lagrange multipliers for the state-constraints. Under natural assumptions, we are able to show the existence of bounded and measurable Lagrange multipliers. The method is based on results from the theory of continuous linear programming problems.

A New Algorithm for State-Constrained Separated Continuous Linear Programs

During the last few decades, significant progress has been made in solving large-scale finite-dimensional and semi-infinite linear programming problems. In contrast, little progress has been made in solving linear programs in infinite-dimensional spaces despite their importance as models in manufacturing and communication systems. Inspired by the research on separated continuous linear programs, we propose a new class of continuous linear programming problems that has a variety of important applications in communications, manufacturing, and urban traffic control. This class of continuous linear programs contains the separated continuous linear programs as a subclass. Using ideas from quadratic programming, we propose an efficient algorithm for solving large-scale problems in this new class under mild assumptions on the form of the problem data. We prove algorithmically the absence of a duality gap for this class of problems without any boundedness assumptions on the solution set. We show this class of problems admits piecewise constant optimal control when the optimal solution exists. We give conditions for the existence of an optimal solution. We also report computational results which illustrate that the new algorithm is effective in solving large-scale realistic problems (with several hundred continuous variables) arising in manufacturing systems.