Mathematical Programming Problem Research Articles

Co-optimization of thermal power plant flowchart, thermodynamic cycle parameters, and design parameters of components

The improvement in the efficiency of an energy plant depends on a rational development of its flowchart and choice of parameters along with the load schedule, equipment reliability, operating mode, etc. It is advisable to study such complex technical systems with the methods of mathematical modeling and optimization. The paper presents an approach to the development of optimal flowcharts and selection of parameters of energy plants. The approach is based on the combination of a method for optimization of the most complex flowchart and a method for solving discrete-continuous problems of nonlinear mathematical programming. A case study of the co-optimization of design parameters, operating parameters and equipment mix for the integrated gasification combined-cycle plant is demonstrated. The optimization calculations were carried out by the criterion of the minimum price of electricity for a given internal rate of return on investment and the maximum energy efficiency of the plant. Several optimal solutions meeting the different criteria are obtained. The proposed approach can be used for optimization of flowcharts and parameters of other complicated energy plants (high-efficiency combined-cycle plants, ultra-supercritical steam cycles, integrated power plants for electricity and synthetic liquid fuel co-production from coal, etc.).

Optimality Conditions for Nonconvex Nonsmooth Optimization via Global Derivatives

The notions of upper and lower global directional derivatives are introduced for dealing with nonconvex and nonsmooth optimization problems. We provide calculus rules and monotonicity properties for these notions. As a consequence, new formulas for the Dini directional derivatives, radial epiderivatives and generalized asymptotic functions are given in terms of the upper and lower global directional derivatives. Furthermore, a mean value theorem, which extend the well-known Diewert’s mean value theorem for radially upper and lower semicontinuous functions, is established. We also provide necessary and sufficient optimality conditions for a point to be a local and/or global solution for the nonconvex minimization problem. Finally, applications for nonconvex and nonsmooth mathematical programming problems are also presented.

Sustainable Waste Management Drilling Process in Fuzzy Environment

Sustainable management issues of waste during drilling oil wells in marine conditions, the process of disposal of drill cuttings in the conditions of deficiency, and fuzzy initial information using fuzzy inference system are investigated. Based on the conducted system analysis, the main criteria for controlling the process of re-injection of suspended drill cuttings were analyzed and selected. We described the technology of preparation and injection of drill cuttings slurry into the underground horizon. The method of modeling and management of the process of disposal of drilling cuttings in the marine environment in a fuzzy environment with the use of fuzzy inference system, which helps to overcome the problems of scarcity and fuzziness of the original information due to the knowledge and experience of experts are proposed. The scheme and structure of the elements of the fuzzy inference system based on the Mamdani algorithm are given. The implementation of the fuzzy output system procedure was carried out in MatLab using Fuzzy Logic Toolbox. For the purpose of sustainable waste management in the process of oil production of marine fields, waste management tasks are formulated as a fuzzy mathematical programming problem, which takes into account economic and environmental criteria and many production constraints that may be fuzzy. Since the vector of such criteria is characterized by inconsistency, the developed methods for solving the set tasks of sustainable management are based on various tradeoff schemes modified to work in a fuzzy environment. The novelty and originality of the developed methods lies in the fact that, unlike the well-known methods of similar methods for solving fuzzy problems, they are set and solved without conversion to a system of equivalent deterministic problems, with-out losing the main part of the collected fuzzy information. This allows, through the full use of the original fuzzy information, to obtain a more adequate solution to the fuzzy problem of the real problem under production conditions.

DYNAMIC PROBLEM OF FORMATION OF SECURITIES PORTFOLIO UNDER UNCERTAINTY CONDITIONS

The analysis of known methods for solving the problem of forming a portfolio of securities in the face of uncertainty is carried out. Traditionally, the problem is solved under the assumption that for each type of asset, the values of the main statistical characteristics of the random value of their profitability (mathematical expectation and variance) are known. At the same time, the variance of portfolio returns, which is minimized, is used as a criterion for portfolio optimization. Two alternative approaches to solving the formulated problem are proposed. The first of them provides a decision on the criterion of the probability that the random total portfolio return will not be lower than the given. It is assumed that the random return for each type of asset is distributed normally and the statistical characteristics of the respective densities are known. The original problem is reduced to the problem of maximizing the quadratic fractional criterion in the presence of linear constraints. To solve this non-standard optimization problem, a special iterative algorithm is proposed that implements the procedure for sequential improvement of the plan. The method converges and the computational procedure for obtaining a solution can be stopped by any of the standard criteria. The second approach considers the possibility of solving the problem under the assumption that the distribution densities of random asset returns are not known, however, based on the results of preliminary statistical processing of the initial data, estimates of the values of the main numerical characteristics for each of the assets are obtained. To solve the problem, a new mathematical apparatus is used – continuous linear programming, which is a generalization of ordinary linear programming to the case when the task variables are continuous. This method, in the considered problem, is based on solving an auxiliary problem: finding the worst-case distribution density of a random total portfolio return at which this total return does not reach an acceptable threshold with maximum probability. Now the main minimax problem is being solved: the formation of the best portfolio in the worst conditions. The resulting computational scheme leads to the problem of quadratic mathematical programming in the presence of linear constraints. Next, a method is proposed for solving the problem of forming a portfolio of securities, taking into account the real dynamics of the value of assets. The problem that arises in this case is formulated and solved in terms of the general theory of control, using the Riccati equation.

Synthesis of the Topological Structure of Distributed Terminal System for Audio Monitoring of Users of Local Information Spaces

A widespread use of multi-user interfaces, due to multimodality of traditional interpersonal communication, a transition to a polymerized presentation of information and systems, has allowed the creation of new approaches to their implementation based on distributed terminal systems. An approach to the synthesis of topological structures of such systems implemented in two stages is proposed in the article. The first stage determines a minimum set of communication nodes and their location based on the requirements for the availability of communication nodes for various categories of users and the globality of a distributed terminal system. The second stage determines options for constructing communication nodes and connections between them, which ensure the performance of audio monitoring functions of users of local information spaces while ensuring continuity of a bridge for different categories of users. A model example of the synthesis of a distributed terminal system for audio monitoring of two categories of users (adults and children) in the local information space (home), voice control subsystems of the "smart home" is presented. As a part of its solution, at each stage of the synthesis, the initial data are determined, a formal formulation of the synthesis problem is carried out, an algorithm for the solution and the results are presented. So the task of the first stage of the synthesis is a linear integer mathematical programming problem, solved in the model example by the simplex method, the solution of the second stage problem is based on the alternative graph formalization and the method of "branches and borders". The obtained results clearly demonstrate the capabilities of the proposed scientific and methodological tools for the synthesis of the topological structure of distributed terminal systems and the prospects of its use in the newly arising tasks of the technical implementation of new infocommunication technologies and services.

New concepts and an algorithm for multiobjective bilevel programming: optimistic, pessimistic and moderate solutions

Bilevel optimization deals with hierarchical mathematical programming problems in which two decision makers, the leader and the follower, control different sets of variables and have their own objective functions subject to interdependent constraints. Whenever multiple objective functions exist at the lower-level problem, the leader should cope with the uncertainty pertaining to the follower’s reaction. The leader can adopt a more optimistic or more pessimistic stance regarding the follower’s choice within his efficient region, which is restricted by the leader’s choice. Moreover, the leader may also have multiple objective functions. This paper presents new concepts associated with solutions to problems with multiple objective functions at the lower-level and a single or multiple objective functions at the upper-level, exploring the optimistic and pessimistic leader’s perspectives and their interplay with the follower’s choices. Extreme solutions (called optimistic/deceiving and pessimistic/rewarding) and a moderate solution, resulting from the risk the leader is willing to accept, are defined for problems with a single objective at the upper-level (semivectorial problems). Definitions of optimistic and pessimistic Pareto fronts are proposed for problems with multiple objective functions at the upper-level. These novel concepts are illustrated emphasizing the difficulties associated with the computation of those solutions. In addition, a differential evolution algorithm, approximating the extreme and moderate solutions for the semivectorial problem, is presented. Illustrative results of this algorithm further stress the challenges and pitfalls associated with the computation and interpretation of results in this kind of problems, which have not been properly addressed in literature and may lead to misleading conclusions.

Characterization of generalized FJ and KKT conditions in nonsmooth nonconvex optimization

In this paper, we investigate new generalizations of Fritz John (FJ) and Karush–Kuhn–Tucker (KKT) optimality conditions for nonconvex nonsmooth mathematical programming problems with inequality constraints and a geometric constraint set. After defining generalized FJ and KKT conditions, we provide some alternative-type characterizations for them. We present characterizations of KKT optimality conditions without assuming traditional Constraint Qualification (CQ), invoking strong duality for a sublinear approximation of the problem in question. Such characterizations will be helpful when traditional CQs fail. We present the results with more details for a problem with a single-inequality constraint, and address an application of the derived results in mathematical programming problems with equilibrium constraints. The objective function and constraint functions of the dealt with problem are nonsmooth and we establish our results in terms of the Clarke generalized directional derivatives and generalized gradient. The results of the current paper cover classic optimality conditions existing in the literature and extend the outcomes of Flores-Bazan and Mastroeni (SIAM J Optim 25:647–676, 2015).

Mathematical Apparatus of Optimal Decision-Making Based on Vector Optimization

We present a problem of “acceptance of an optimal solution” as a mathematical model in the form of a vector problem of mathematical programming. For the solution of such a class of problems, we show the theory of vector optimization as a mathematical apparatus of acceptance of optimal solutions. Methods of solution of vector problems are directed to problem solving with equivalent criteria and with the given priority of a criterion. Following our research, the analysis and problem definition of decision making under the conditions of certainty and uncertainty are presented. We show the transformation of a mathematical model under the conditions of uncertainty into a model under the conditions of certainty. We present problems of acceptance of an optimal solution under the conditions of uncertainty with data that are represented by up to four parameters, and also show geometrical interpretation of results of the decision. Each numerical example includes input data (requirement specification) for modeling, transformation of a mathematical model under the conditions of uncertainty into a model under the conditions of certainty, making optimal decisions with equivalent criteria (solving a numerical model), and, making an optimal decision with a given priority criterion.

Error Bound property in mathematical programming

This article is devoted to the Error Bound property (also named R-regularity) in mathematical programming problems. This property plays an important role in analyzing the convergence of numerical optimization algorithms, a topic covered by multiple publications, and at the same time it is a relatively generic constraint qualification that guarantees the satisfaction of the necessary Kuhn – Tucker optimality conditions in mathematical programming problems. In the article, new sufficient conditions for the error bound property are described, and it’s also shown that several known necessary conditions are insufficient. The sufficient conditions obtained can be used to prove the regularity of a large class of sets including sets that cannot be proven regular by other known constraints.

Preface

The ConferenceOptimization is a process to achieve the ideal or optimal value of an objective function. In the discipline of mathematics, optimization also refers to the study of problems trying to find the minimum and maximum values of a real function. To achieve these minimum and maximum values, some processes were systematically carried out by selecting integers or real numbers that will provide optimal solutions.The true discriminant between “easy” and “hard” optimization problems in mathematical programming is the convex/non-convex issue. A problem is convex if it is a minimization of a convex function (or a maximization of a concave function) where the admissible points are in a convex set. The fundamental result in convex analysis says that a locally optimal solution of a convex problem is also globally optimal. This is practically very important since algorithmic termination tests occur at each iteration in the algorithm, they must be computationally very efficient; thus, virtually all termination conditions of optimization algorithms are local conditions, i.e. they test whether the current solution is locally optimal with respect to a pre-defined neighborhood. If the problem is convex this is enough to guarantee that the solution is globally optimal. Non-convex problems, on the other hand, may have many different local optimal, and choosing the best one is an extremely hard task.The field of applied mathematics that studies extremal locations of non-convex functions subject to (possibly) non-convex constraints is called Global Optimization. Research on global optimization has been continuing rapidly and many of the findings are of new techniques that are more sophisticated and efficient.The 7th International Conference on Global Optimization and Its Application (ICoGOIA) 2018 is hosted and organized by Faculty of Mathematics and Natural Sciences Universitas Pakuan, Indonesia.List of Conference Publication, Conference Theme, Organization, Participants, Committee, Secretariat, Welcoming Speech, Table of Contents and all the titles are available in this pdf.

Computer simulation of sublayer temperature mode during deposition of diamond coatings using gas-plasma method

The research presents a computer simulation of the temperature field on the sublayer surface, performed in a stationary mode during the deposition of diamond coatings using the gas-plasma method with different parameters of the cooling system. The optimization problem of mathematical programming with the search for the extremum of the performance function by varying the controlled parameters within the permissible range is considered.

Rigid block modelling of historic masonry structures using mathematical programming: a unified formulation for non-linear time history, static pushover and limit equilibrium analysis

A unified formulation is presented for non-linear time history, static pushover and limit analysis of historic masonry structures modelled as 2D assemblages of rigid blocks interacting at no-tension, frictional contact interfaces. The dynamic, incremental static and limit analysis problems are formulated as mathematical programming problems which are equivalent to the equations system governing equilibrium, kinematics and contact failure. Available algorithms from the field of mathematical programming, contact dynamics and limit analysis are used to tackle the contact problems between rigid blocks in a unified framework. To evaluate the accuracy and computational efficiency of the implemented formulation, applications to numerical case studies from the literature are presented. The case studies comprise rigid blocks under earthquake excitation, varying lateral static loads and sliding motion. A set of two leaves wall panels and an arch-pillars system are also analysed to compare failure mechanisms, displacement capacity and magnitudes of lateral loads promoting the collapse.

Reduction of multiple‐input multiple‐output specific absorption rate synthesized using a fast estimation model

A technique that improves multiple-input multiple-output (MIMO) specific absorption rate (SAR) is proposed for multiple-antenna transmitters. When multiple antennas are implemented, the SAR evaluation is more complex than the stand-alone counterpart. Techniques for the reduction of stand-alone SAR cannot be applied to the MIMO scenario straightforwardly. To reduce MIMO SAR and preserve radiation characteristics simultaneously, a technique based on solving the fast estimation model of MIMO SAR is proposed. The proposed technique is cast into two phases. In the phase one, the fast estimation model is constructed for MIMO SAR. This model can predict arbitrary combinations of MIMO SAR once the contribution of the electric field of each antenna is obtained. The capability of the prediction is validated through 19 scenarios. In the phase two, the fast estimation model is formulated as a mathematical programming problem, where the current amplitude of each antenna is treated as decision variables. The combination of the decision variables that leads to the lowest SAR is thus the optimum solution. The proposed technique is tested using a quad-element antenna system that operates in 3.4 to 3.7 GHz. The results indicate that most of the scenarios depict reduced SAR. The averaging MIMO SAR is improved by 13.75%.

On Semi-infinite Mathematical Programming Problems with Equilibrium Constraints Using Generalized Convexity

In this paper, we consider semi-infinite mathematical programming problems with equilibrium constraints (SIMPPEC). By using the notion of convexificators, we establish sufficient optimality conditions for the SIMPPEC. We formulate Wolfe and Mond–Weir-type dual models for the SIMPPEC under the invexity and generalized invexity assumptions. Weak and strong duality theorems are established to relate the SIMPPEC and two dual programs in the framework of convexificators.

Metric Regularity of Quasidifferentiable Mappings and Optimality Conditions for Nonsmooth Mathematical Programming Problems

This article is devoted to the analysis of necessary and/or sufficient conditions for metric regularity in terms of Demyanov-Rubinov-Polyakova quasidifferentials. We obtain new necessary and sufficient conditions for the local metric regularity of a multifunction in terms of quasidifferentials of the distance function to this multifunction. We also propose a new MFCQ-type constraint qualification for a parametric system of quasidifferentiable equality and inequality constraints and prove that it ensures the metric regularity of a multifunction associated with this system. As an application, we utilize our constraint qualification to strengthen existing optimality conditions for quasidifferentiable programming problems with equality and inequality constraints. We also prove the independence of the optimality conditions of the choice of quasidifferentials and present a simple example in which the optimality conditions in terms of quasidifferentials detect the non-optimality of a given point, while optimality conditions in terms of various subdifferentials fail to disqualify this point as non-optimal.

Real-Time Integrated Holding and Priority Control Strategy for Transit Systems

This paper presents a real-time integrated holding and priority control strategy for bus rapid transit (BRT) and high-frequency segregated transit systems, in a network approach (considering the whole transit circuit). The integrated control strategy implements simultaneously the required bus headway corrections and the bus priority through signalized intersections with the objective of minimizing the total delay of passengers that are onboard and at stops. The developed model, in the form of a mathematical programming problem, presents peculiarities for describing the behavior of BRT and transit systems such as bus capacity, differentiation between passenger boarding and alighting processes in single stops, terminals and stations, and bus priority at signalized intersections. The iterative procedure proposed for solving the optimization problem brings about simplicity and efficiency for real-time applications. The results obtained by simulation for a transit scenario of the trunk line 10 of the city of Blumenau, Santa Catarina, Brazil, show the efficiency of the proposed strategy and potential of practical application.

Co-evolutionary algorithms to solve hierarchized Steiner tree problems in telecommunication networks

This study investigates a hierarchized Steiner tree problem in telecommunication networks. In such networks, users must be connected to capacitated hubs. Additionally, selected hubs must be connected to each other and to extra hubs, if necessary, by considering the latency of the resultant network. A connection between hubs can be considered to be a Steiner tree. This Steiner tree problem is modeled as a bi-level mathematical programming problem that considers two decision levels. In the upper-level, the allocation of users to hubs is performed to minimize the total network connection cost. The lower-level minimizes the user latency concerning the information that flows through the capacitated hubs. Further, two co-evolutionary schemes are developed to solve this bi-level model. The first scheme is an individual–population approach, whereas the second scheme is the traditional population–population approach. The first proposed algorithm exploits the structure of the problem by employing parallel computing in one of the populations. Numerical results depict the effectiveness of the proposed algorithms when the lower-level problem cannot be optimally solved efficiently. Furthermore, the advantages of the proposed schemes over an evolutionary one are exhibited. Finally, the hybridization of both co-evolutionary schemes is implemented to improve the semi-feasible solutions obtained by the second scheme, showing its effectiveness to solve the problem.

Multi-component energy modeling and optimization for sustainable dry gear hobbing

Sustainable machining becomes a key priority for manufacturing industries due to the ever growing energy costs, associated environmental impacts and carbon emissions. As one of the frequent activities in metal machining, dry gear hobbing contributes to a significant portion of energy consumption. Process parameter optimization is an effective method of decreasing energy from process control perspective. However, hobbing parameter optimization is rarely involved in previous studies. To this end, a multi-component energy model is first developed on a basis of energy characteristics analysis of dry gear hobbing machines. Then, the optimization of hobbing parameters for the minimizing energy consumption and production cost is formulated as mathematical programming problem with a systematic consideration of machining constraints. Finally, the optimization problem is solved by a modified multi-objective imperialist competitive algorithm (MOICA). The results demonstrate that the energy-efficient gear hobbing can be achieved through a collaborative effort of predictive modeling and parameter optimization.

Exploiting Lagrange duality for topology optimizationwith frictionless unilateral contact

This paper presents tractable reformulations of topology optimization problems of structures subject to frictionless unilateral contact conditions. Specifically, weconsider stiffness maximization problems of trusses and continua. Based on the Lagrange duality theory, we derive formulations that do not involve complementarity constraints. It is often that a structural optimization problem with contact conditions is formulated as a mathematical programming problem with complementarityconstraints (MPCC problem). However, MPCC usually requires special treatment for numerical solution, because it does not satisfy standard constraint qualifications. In contrast, to the formulation presented in this paper, we can apply standard optimization approaches. Numerical experiments on trusses and continua are performed to examine efficiency of the proposed approach.

Geometrical synthesis of four-bar gear train with related gears

The method of optimal geometric synthesis of four-bar cylindrical gear with related gears with one external and one internal gearing is proposed. The method is based on a new calculation system that increases the number of synthesis output parameters to six, which makes it possible to considerably expand the domain of existence of the synthesis problem solution with the simultaneous synthesis of two gearing. The software implementing the proposed technique is described. To solve the problem of nonlinear mathematical programming the library of Optimization Toolbox functions included in Matlab is used. Examples of calculations are given.