Mathematical Programming Problem Research Articles

On the Problem of Maximizing the Probability of Successful Passing of a Time-Limited Test

—The problem of finding the optimal sequence of performing a set of tasks in a timelimited test is considered. That is, a task group is allocated for mandatory initial execution in the test, the remaining tasks are performed during the remaining time until the end of the test. For each correctly completed task of the test, the subject is awarded a certain number of points. The proposed criterion is the probability that the total number of points scored for the test exceeds a certain level, which is a fixed parameter, while simultaneously fulfilling the time limit of the test. Random parameters are the user’s response time to each test task. The correctness of the user’s answer to the task is modeled by a random variable with a Bernoulli distribution. The resulting stochastic bilinear programming problem boils down to a deterministic integer problem of mathematical programming.

A class of constrained optimal control problems arising in an immunotherapy cancer remission process

<p>By considering both the single drug dose and the total drug input during the treatment period, we propose a new optimal control problem by maximizing the immune cell levels and minimizing the tumor cell count, as well as the negative effects of the total drug quantity over time. To solve this problem, the control parameterization technique is employed to approximate the control function by a piecewise constant function, which gives rise to a sequence of mathematical programming problems. Then, we derive gradients of the cost function and/or the constraints in the resulting problems. On the basis of this gradient information, we develop a numerical approach to seek the optimal control strategy for a discrete drug administration. Finally, numerical simulations are conducted to assess the impact of the total drug input on the tumor treatment and to evaluate the rationality of the treatment strategy within the anti-cancer cycle. These results provide a theoretical framework that can guide clinical trials in immunotherapy.</p>

СРАВНЕНИЕ МЕТОДОВ РЕШЕНИЯ ТРАНСПОРТНОЙ ЗАДАЧИ

Mathematical programming is a mathematical discipline that studies various extremal problems and develops algorithms for solving them. Among the problems are those of mathematical integer programming. Integer programming is indispensable for solving mathematical programming problems in which some or all of the variables take on integer values. The problems include the transport problem. The paper considers the transport problem and three linear programming methods for solving it: the method of potentials, the method of differential rents and the simplex method. The paper analyzes and assesses the effectiveness of the three algorithms for optimizing transport problems. Several linear optimization algorithms have been studied and presented. The paper contains the basic concepts, formulas, and algorithms of the methods used. The method of potentials, the method of differential rents and the simplex method are considered. The results of solving the problem using these methods are presented and compared. All calculations were performed using Excel.

Optimizing Chance Constraint Multiple-Objective Fractional Mathematical Programming Problem Involving Dependent Random Variable

This manuscript suggests a methodology to solve chance constraint multiple-objective linear fractional mathematical programming problem in which the parameters are dependent random variables to each other. The proposed problem is formulated by taking few of the parameters as continuous dependent random variables. The proposed model cannot be solved directly by using existing methodology. Thus in order to solve the proposed model, an equivalent deterministic model is derived. The procedure to solve the proposed model is accomplished in two main steps. Initially, the proposed multiple-objective chance constraint linear fractional mathematical problem is transformed to deterministic equivalent multiple-objective linear fractional mathematical programming by the help of chance constrained method. In the second step, multiple-objective functions, which consist of fractional functions is solved by using lexicographic programming approach. Finally, an example is mentioned to illustrate the methodology.

Lua APIs for mathematical optimization

In this paper we present our Lua programming language libraries that enable modeling and solving mixed integer linear and nonlinear mathematical optimization problems. On one side, these libraries provide a framework for developing algorithms written in Lua that use certain well known solvers within more complex procedures. On the other side, they facilitate the transformation of both input and output data of a mathematical programming problem, in order to compute the standardized coefficients that have to be transmitted to the solvers or prepare solution of solved problems for further data processing. Both of these use cases, together with Lua programming language simplicity, versatility and performance, make Lua a suitable programming language for use in scientific and engineering researches.

Optimality conditions for nonconvex mathematical programming problems using weak subdifferentials and augmented normal cones

Optimality conditions for nonconvex mathematical programming problems using weak subdifferentials and augmented normal cones

Methodology of delivering space parts, tools and accessories to a near Earth orbital station on the basis of the reliability factor

The main aspects of the deployment of multi-module orbital manned stations are considered. The methodology for planning the delivery of spare parts, tools and accessories used in planning of transport operations to the International Space Station, is analyzed. It is shown that its application is related to the problem of mathematical programming with a large number of variables, which complicates the process of finding the optimal planning option in terms of economic efficiency. Transport operations to national manned orbital stations are carried out less frequently as compared to the International Space Station due to the enormous costs of spacecraft launches taking into account possible emergency situations. An approach to improving the methodology for planning the delivery of spare parts, tools and accessories for possible use on national manned orbital stations is proposed. It consists in taking into account the reliability of on-board equipment. The advantage of the improved methodology is simplification of planning transport operations without the use of mathematical programming. In addition, the improved methodology is clear, simple and can be implemented step by step.

Modern algorithm of linear and quadratic programming in optimization and problems of deformation of structures of variable structure in contact conditions

Various mathematical problems, in which the goal of finding the extremum of a functional is set, belong to the problems of mathematical programming and optimization problems. Practically directed problems of finding the optimal solution are extremely numerous in economics, management, technology, and others. They are related to increasing production efficiency, reducing resource costs, improving design solutions and technological processes, reducing mass, dimensions, etc. Among them, an important role is given to the methods of limiting the maximum stresses caused by external loads. Solving such problems begins with mathematical formalization. Constructive, economic or technological indicators are chosen as variation parameters. The search for the best solution is reduced to the selection of a set of parameters that provide a stationary value of the objective function. Extreme problems of practical orientation contain equality-inequality constraints in mathematical models. In improving the technical characteristics of machines, a significant role belongs to engineering and technical workers, who find optimal options at the design stage. At the same time, an essential element of the design process is the modeling of the determining processes in structures, taking into account the main influencing factors and behavior scenarios. Optimization is an important area of applied mathematics that provides effective tools for such modeling. Universal Algorithm is proposed in work [3] – a numerous scheme for solving quadratic programming (QP) problems for calculating the optimal point of a wide range of applied problems. At the same time, the linear programming (LP) problem is considered as a partial case of the (QP) problem. That is, in the universal algorithm for setting 2 optimization problems, they are formalized in a single and convenient form of symmetric matrix dependence, which makes it possible to build a single effective algorithm based on matrix algebra operations. In particular, it allows you to consider the practical tasks of calculating VAT in constructions of a variable structure consisting of separate parts connected by one-way connections. The main goal of this work is to analyze the behavior of the algorithm when increasing the number of constraints of the inequality type, to refine the computational scheme, and to formulate conclusions. Two model problems are considered as examples of the algorithm. This is a classic "transport" problem of LP and the behavior of a model of a bridge structure with one-way connections in cables under variations of wind loads. The number of ropes has been increased to 20, and the limits of one-way connections to 40.

Fuzzy Programming Approach to Solve Multi-Objective Fully Fuzzy Transportation Problem

The aim of this study is presenting the solution methodology of multi-objective fuzzy transportation problem with fuzzy decision variables, where all the input parameters and decisions variables of the programming problems are assumed to be triangular fuzzy number and triangular fuzzy decision variables respectively. Moreover the objectives under considerations are minimization of cost of transportation and minimization of shipping time under fuzzy environment. The fuzziness of the objective functions and the fuzzy constraints of the programming problem are defuzzified using the ranking function and the equality property between two fuzzy numbers, respectively. The consequent crisp multi-objective fuzzy transportation problem is tackled by employing fuzzy mathematical programming approach. Finally fuzzy decision is made after solving the resultant mathematical programming problem using LINGO(Schrage and LINDO Systems (1997)) software. Illustrative numerical example is presented in support of the proposed methodology.

Adaptive Subgradient Methods for Mathematical Programming Problems with Quasiconvex Functions

Adaptive Subgradient Methods for Mathematical Programming Problems with Quasiconvex Functions

Controlling the Process of Formation of the Scientific and Technological Background in Aircraft Construction Based on the Assessment of a Deficit of Functional Properties of Promising Aircraft

The existing experience of the design of aviation technology proves that in a number of cases a promising aircraft (AC) that satisfies the set of requirements cannot be created within the framework of the accumulated scientific and technical groundwork in the field of AC manufacturing. As a result, there is a need to determine priority scientific and technical solutions, whose implementation, under time and financial constraints, will ensure the creation of an AC that meets the stated requirements. It is shown that such a problem can be interpreted as a so-called generalized optimization problem, whose difference from the classical mathematical programming problem lies in the fact that the set of permissible values of flight performance characteristics (FPC) is an additional object of optimization. A constructive algorithm for solving such a problem is proposed, whose capabilities are illustrated using the example of designing a transport ramp aircraft

UPPER BOUND FOR THE COMPETITIVE FACILITY LOCATION PROBLEM WITH DEMAND UNCERTAINTY

We consider a competitive facility location problem with two competing parties operating in a situation of uncertain demand scenario. The problem to find the best solutions for the parties is formulated as a discrete bi-level mathematical programming problem. In the paper, we suggest a procedure to compute an upper bound for the objective function on subsets. The procedure could be employed in implicit enumeration schemes capable to compute an optimal solution for the problem under study. Within the procedure, additional constraints iteratively augment the high-point relaxation of the initial bi-level problem, what strengthens the relaxation and improves the upper bound’s quality. New procedure to generate such cuts allows to construct the strongest cuts without enumerating the parameters encoding them.

Robust Solution of the Multi-Model Singular Linear-Quadratic Optimal Control Problem: Regularization Approach

We consider a finite horizon multi-model linear-quadratic optimal control problem. For this problem, we treat the case where the problem’s functional does not contain a control function. The latter means that the problem under consideration is a singular optimal control problem. To solve this problem, we associate it with a new optimal control problem for the same multi-model system. The functional in this new problem is the sum of the original functional and an integral of the square of the Euclidean norm of the vector-valued control with a small positive weighting coefficient. Thus, the new problem is regular. Moreover, it is a multi-model cheap control problem. Using the solvability conditions (Robust Maximum Principle), the solution of this cheap control problem is reduced to the solution of the following three problems: (i) a terminal-value problem for an extended matrix Riccati type differential equation; (ii) an initial-value problem for an extended vector linear differential equation; (iii) a nonlinear optimization (mathematical programming) problem. We analyze an asymptotic behavior of these problems. Using this asymptotic analysis, we design the minimizing sequence of state-feedback controls for the original multi-model singular optimal control problem, and obtain the infimum of the functional of this problem. We illustrate the theoretical results with an academic example.

Determining the area of a technical system stability in fuzzy sets terms

Ensuring the stability of the technical systems functioning, including those in water transport, is an important area for the application of mathematical methods, including probabilistic and mathematical-statistical methods. The stochastic nature of disturbances affecting complex systems and worsening the level of stability of the latter should be reflected in the implemented human-machine control loops. One of the directions here is the accounting and use of (usually competing) expert assessments, on the one hand, and a reasonable definition of stability indicators, a choice from a variety of possibilities, on the other hand. From a general point of view of management theory, the method of using expert assessments developed in the paper can be considered as an integral part of the risk management problem that is being developed so far, mainly for economic systems and using only financial criteria. The concept of the minimum allowable efficiency of a technical system is introduced in the paper. On its basis, an approach in which the indicator of the stability of a technical system is linked to a system-wide criterion indicator of the quality of its functioning is implemented. The corresponding mathematical model as a problem of mathemati- cal programming is formulated. Approaches to its solution including for the nonconvex multiextremal case are discussed. A feature of the proposed approach to the definition of the integral indicator of the stability of the dynamics of a technical system is the use of expert assessments in determining the probable range of changes in the controlled variables of the optimization model, implemented as a mathematical programming problem, which logically leads to the use of fuzzy set theory. With the non-linearity of the objective functions and restrictions that arise in this case, the gradient method of searching for a conditional optimum is effective.

On Frèchet normal cone for nonsmooth mathematical programming problems with switching constraints

This paper is devoted to the study of a class of nonsmooth programming problems with switching constraints (abbreviated as, (NMPSC)), where all the involved functions in the switching constraints are assumed to be locally Lipschitz. We investigate the properties of Frèchet normal cone of (NMPSC). In particular, we introduce two Guignard type constraint qualifications for (NMPSC) in terms of Michel-Penot subdifferential. Moreover, we derive two estimates for the Frèchet normal cone of (NMPSC) and further establish stationarity conditions at an optimal solution for (NMPSC). To the best of our knowledge, this is for the first time Frèchet normal cone for (NMPSC) have been studied in the setting of Euclidean spaces.

Limit load and failure mechanisms of a vertical Hoek-Brown rock slope

The problem considered in this short note is the limit load determination of a vertical rock slope. The classical limit theorem is employed with the use of adaptive finite elements and nonlinear programming to determine upper and lower bound limit loads of a Hoek-Brown vertical rock slope. The objective function of the mathematical programming problem is such as to optimize a boundary load, which is known as the limit load, resembling the ultimate bearing capacity of a strip footing. While focusing on the vertical slope, parametric studies are carried out for several dimensionless ratios such as the dimensionless footing distance ratio, the dimensionless height ratio, and the dimensionless rock strength ratio. A comprehensive set of design charts is presented, and failure envelopes shown with the results explained in terms of three identified failure mechanisms, i.e. the face, the toe, and the Prandtl-type failures. These novel results can be used with great confidence in design practice, in particularly noting that the current industry-based design procedures for the presented problem are rarely found.

Collaborative capacity planning method of wind-photovoltaic-storage equipment in microgrids considering different energy selling incomes

A microgrid is a promising small-scale power generation and distribution system. The selling prices of wind turbine equipment (WT), photovoltaic generation equipment (PV), and battery energy storage equipment (BES) have a significant impact on microgrid profits, which, in turn, affects the planning capacity of renewable energy. However, existing research has not yet conducted in-depth modeling and analysis for different kinds of energy generation electricity prices. This paper proposes an optimal capacity planning method for wind-photovoltaic-storage equipment, considering different energy selling incomes in microgrids. Stochastic characteristics of renewable energy (WT and PV), selling prices of different types of energy, and timing coupling characteristic are considered in the proposed model. In addition, the configuration capacities of WT, PV, and BES are modeled as discrete decision variables, according to the type of specific equipment. The comprehensive life cycle cost (LCC) is considered an objective function. It can be found that the proposed collaborative capacity planning model is a mathematical programming problem with complex nonlinear constraints and integer variables. To solve this problem, a cultural gray wolf optimization algorithm (CGWO) is applied in this paper. The proposed method’s efficiency, convergence, superiority, and effectiveness are verified through a case study. Moreover, the impact of different new energy sales prices on capacity planning results is also revealed in the article.

On the structure of optimal solutions in a mathematical programming problem in a convex space

We consider an optimization problem in a convex space E with an affine objective function, subject to J affine constraints, where J is a given nonnegative integer. We apply the Feinberg-Shwartz lemma in finite dimensional convex analysis to show that there exists an optimal solution, which is in the form of a convex combination of no more than J+1 extreme points of E. The concerned problem does not seem to fit into the framework of standard convex optimization problems.

Criterion of solvability of the discriminant analysis problem underlying the medical organization management model

The article considers the task of building an effective and efficient organization of the management processes of a medical organization. A distinctive feature is noted - social orientation as the main function, and not just profit maximization with limited resources. This approach requires taking into account not only financial and resource components, but also the impact of many other external factors. However, the authors, realizing the difficulty of describing them, propose to take into account the effect of the "social orientation of the institution" in an implicit form when describing the financial condition of the organization. Taking into account implicit, including external, factors is possible when implementing models not only of statistical and financial analysis, as is usually accepted, but also their extensions to some image recognition methods, which often have a simple computer implementation. In particular, the formalization of the financial condition model of the organization is proposed, which reduces to the problem of mathematical programming with state vectors of both internal and external factors. The possibility of considering a decision-making model that would take into account the influence of possible external factors is also considered, while the "informality" of the problem in the sense of the absence of classical methods of solution is indicated. In this regard, it is proposed to reduce the model to the task of discriminant analysis, which is essentially a system of inequalities. Methods for solving systems of linear inequalities are known, but they are difficult to implement. For this reason, it is proposed to add an optimality criterion (objective function) to the discriminant analysis model, which allows to obtain a linear programming problem that already has simpler computer implementations of solutions. However, this possibility is present only in the case of linear constraints, which is a somewhat special case of the general problem. To solve the problem, in its broadest sense, it is proposed not only to reduce it to mathematical programming, and then to discriminant analysis, but also an effective recognition algorithm indicating the solvability of the problem. The process of constructive problem solving often requires a lot of time and effort, even with the powerful development of modern technologies. With the help of the given recognition algorithm, the discriminant analysis model in some formulation is output in a matrix form, which allows us to conclude when the model has the existence of a separating functional. The authors also propose the use of majority committee methods. The derived algorithm is quite simple to implement, but at the same time effective and efficient in practice, which makes it universal for this kind of tasks.

Optimality Conditions in DC-Constrained Mathematical Programming Problems

This paper provides necessary and sufficient optimality conditions for abstract-constrained mathematical programming problems in locally convex spaces under new qualification conditions. Our approach exploits the geometrical properties of certain mappings, in particular their structure as difference of convex functions, and uses techniques of generalized differentiation (subdifferential and coderivative). It turns out that these tools can be used fruitfully out of the scope of Asplund spaces. Applications to infinite, stochastic and semi-definite programming are developed in separate sections.