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Given a linear programming problem with the objective function coefficients vector c and a feasible solution x 0 to this problem, a corresponding inverse linear programming problem is to modify the vector c as little as possible to make x 0 form an optimal solution to the linear programming problem. The modifications can be measured by different distances. In this article, we consider the inverse linear programming problem under the bottleneck-type weighted Hamming distance. We propose an algorithm based on the binary search technique to solve the problem. At each iteration, the algorithm has to solve a linear programming problem. We also consider the inverse minimum cost flow problem as a special case of the inverse linear programming problems and specialize the proposed method for solving this problem in strongly polynomial time. The specialized algorithm solves a shortest path problem at each iteration. It is shown that its complexity is better than the previous one.

A system of interval linear equations is formed and solved.Solutions of system specify the optimal solution set of interval linear programming.Solution space of some solving methods and the optimal solution set are compared.The solutions obtained by proposed method and Monte Carlo method are compared.By solving a large-scale model, the applicability of proposed method is illustrated. Based on the binding constraint indices of the optimal solution to the linear programming (LP) model, a feasible system of linear equations can be formed. Because an interval linear programming (ILP) model is the union of numerous LP models, an interval linear equations system (ILES) can be formed, which is the union of these conventional systems. Hence, a new algorithm is introduced in which an arbitrary characteristic model of the ILP model is chosen and solved. The set of indices of its binding constraints is then obtained. This set is used to form and solve an ILES using the enclosure method. If all the components of the interval solutions to this system are strictly non-negative, the optimal solution set (OSS) of the ILP model is determined as the subscription of the zone created by reversing the signs of the binding constraints of the worst model and the binding constraints of the best model. The solutions to several problems obtained by the new algorithm and a Monte Carlo simulation are compared. The proposed algorithm is applicable to large-scale problems. To this end, an ILP model with 270 constraints and 270 variables is solved.

We present a linear programming based algorithm for a class of optimization problems with a multi-linear objective function and affine constraints. This class of optimization problems has only one objective function, but it can also be viewed as a class of multi-objective optimization problems by decomposing its objective function. The proposed algorithm exploits this idea and solves this class of optimization problems from the viewpoint of multi-objective optimization. The algorithm computes an optimal solution when the number of variables in the multi-linear objective function is two, and an approximate solution when the number of variables is greater than two. A computational study demonstrates that when available computing time is limited the algorithm significantly outperforms well-known convex programming solvers IPOPT and CVXOPT, in terms of both efficiency and solution quality. The optimization problems in this class can be reformulated as second-order cone programs, and, therefore, also be solved by second-order cone programming solvers. This is highly effective for small and medium size instances, but we demonstrate that for large size instances with two variables in the multi-linear objective function the proposed algorithm outperforms a (commercial) second-order cone programming solver.

This paper proposes a bilevel improved fruit fly optimization algorithm (BIFOA) to address the nonlinear bilevel programming problem (NBLPP). Considering the hierarchical nature of the problem, this algorithm is constructed by combining two sole improved fruit fly optimization algorithms. In the proposed algorithm, the lower level problem is treated as a common nonlinear programming problem rather than being transformed into the constraints of the upper level problem. Eventually, 10 test problems are selected involving low-dimensional and high-dimensional problems to evaluate the performance of BIFOA from the aspects of the accuracy and stability of the solutions. The results of extensive numerical experiments and comparisons reveal that the proposed algorithm outperforms the compared algorithms and is significantly better than the methods presented in the literature; the proposed algorithm is an effective and comparable algorithm for NBLPP.

Multi-population algorithms have been widely used for solving the real-world problems. However, it is not easy to get the number of sub-populations to be used for a given problem. This work proposes a self-adaptive multi-population based Jaya (SAMP-Jaya) algorithm for solving the constrained and unconstrained numerical and engineering optimization problems. The Jaya algorithm is a recently proposed advanced optimization algorithm and is not having any algorithmic-specific parameters to be tuned except the common control parameters of population size and the number of iterations. The search mechanism of the Jaya algorithm is upgraded in this paper by using the multi-population search scheme. It uses an adaptive scheme for dividing the population into sub-populations which control the exploration and exploitation rates of the search process based on the problem landscape. The robustness of the proposed SAMP-Jaya algorithm is tested on 15 CEC 2015 unconstrained benchmark problems in addition to 15 unconstrained and 10 constrained standard benchmark problems taken from the literature. The Friedman rank test is conducted in order to compare the performance of the algorithms. It has obtained first rank among six algorithms for 15 CEC 2015 unconstrained problems with the average scores of 1.4 and 1.9 for 10-dimension and 30-dimension problems respectively. Also, the proposed algorithm has obtained first rank for 15 unimodal and multimodal unconstrained benchmark problems with the average scores of 1.7667 and 2.2667 with 50000 and 200000 function evaluations respectively. The performance of the proposed algorithm is further compared with the other latest algorithms such as across neighborhood search (ANS) optimization algorithm, multi-population ensemble of mutation differential evolution (MEMDE), social learning particle swarm optimization algorithm (SL-PSO), competitive swarm optimizer (CSO) and it is found that the performance of the proposed algorithm is better in more than 65% cases. Furthermore, the proposed algorithm is used for solving a case study of the entropy generation minimization of a plate-fin heat exchanger (PFHE). It is found that the number of entropy generation units is reduced by 12.73%, 3.5% and 9.6% using the proposed algorithm as compared to the designs given by genetic algorithm (GA), particle swarm optimization (PSO) and cuckoo search algorithm (CSA) respectively. Thus the computational experiments have proved the effectiveness of the proposed algorithm for solving engineering optimization problems.

In this paper, a wide-neighborhood predictor-corrector feasible interior-point algorithm for linear complementarity problems is proposed. The algorithm is based on using the classical affine scaling direction as a part in a corrector step, not in a predictor step. The convergence analysis of the algorithm is shown, and it is proved that the algorithm has the polynomial complexity $$O\left(\sqrt{n}\log \varepsilon ^{-1}\right)$$ which coincides with the best known iteration bound for this class of mathematical problems. The numerical results indicate the efficiency of the algorithm.

In this paper, we propose an arc-search interior-point algorithm for convex quadratic programming with a wide neighborhood of the central path, which searches the optimizers along the ellipses that approximate the entire central path. The favorable polynomial complexity bound of the algorithm is obtained, namely O(nlog((x0)Ts0/e)) which is as good as the linear programming analogue. Finally, the numerical experiments show that the proposed algorithm is efficient.

For a pair of dual (possibly improper) linear programming problems, a family of matrix corrections is studied that ensure the existence of given solutions to these problems. The case of correcting the coefficient matrix and three cases of correcting an augmented coefficient matrix (obtained by adding the right-hand side vector of the primal problem, the right-hand-side vector of the dual problem, or both vectors) are considered. Necessary and sufficient conditions for the existence of a solution to the indicated problems, its uniqueness is proved, and the form of matrices for the solution with a minimum Euclidean norm is presented. Numerical examples are given.

Nonlinear complementarity and mixed complementarity problems arise in mathematical models describing several applications in Engineering, Economics and different branches of physics. Previously, robust and efficient feasible directions interior point algorithm was presented for nonlinear complementarity problems. In this paper, it is extended to mixed nonlinear complementarity problems. At each iteration, the algorithm finds a feasible direction with respect to the region defined by the inequality conditions, which is also monotonic descent direction for the potential function. Then, an approximate line search along this direction is performed in order to define the next iteration. Global and asymptotic convergence for the algorithm is investigated. The proposed algorithm is tested on several benchmark problems. The results are in good agreement with the asymptotic analysis. Finally, the algorithm is applied to the elastic–plastic torsion problem encountered in the field of Solid Mechanics.

In this paper, we study an optimization problem of minimizing a linear function subject to fuzzy relational inequalities with the addition–min composition. This optimization setting has recently been proposed to model the network cogestion issue when a BitTorrent-like peer-to-peer file-sharing system is used for data transmission. In a 2014 paper, a pseudominimal index (PMI)-based approach was proposed to search for an optimal solution. It turns out that the PMI-based approach may require to solve several to many linear programming problems in order to get an optimal solution. In this paper, we point out that the feasible domain is indeed convex. And we only need to solve a single linear programming problem to generate an optimal solution for the original optimization problem. Furthermore, our approach could be extended to the case with a nonlinear continuous objective function.

This paper deals with a class of mathematical programming problems that includes linear and nonlinear programming problems in a particular form. First, a linear programming problem is considered, and the possibility of deriving its direct complete solution in terms of traditional mathematics without using known iterative computational procedures and algorithms of linear programming, such as the simplex method, is studied. Direct solutions to the problem in the case of minimal dimension with a reduced set of constraints are proposed. It is shown that the derivation of such solutions, as dimension increases, becomes a very complicated problem with increasing dimension and, therefore, is hardly feasible. Some examples of other linear and nonlinear programming problems, which can be obtained from the above-considered problem by means of isomorphic transformations, are presented. The main definitions and preliminary results of tropical mathematics, which are required for the subsequent description and application of tropical optimization methods, are then outlined. A tropical optimization problem is formulated, and direct complete solutions of this problem and of its special cases are given. The above-formulated linear and nonlinear programming problems are reduced to a tropical optimization problem to provide their direct complete solution in terms of tropical mathematics. The solution of the linear programming problem with a reduced set of constraints is written in terms of traditional mathematics.

An algorithm to solve bi-objective quadratic fractional integer programming problems is presented in this paper. The algorithm uses -scalarization technique and a ranking approach of the integer feasible solution to find all nondominated points. In order to avoid solving non-linear integer programming problems during this ranking scheme, the existence of a linear or a linear fractional function is established, which acts as a lower bound on the values of first objective function of the bi-objective problem over the entire feasible set. Numerical examples are also presented in support of the theory.

This article considers bilevel linear programming problems where the coefficients of the objective functions and the constraints in the problem are given as fuzzy parameters. Stackelberg solutions under fuzziness are defined by incorporating the notions of possibility theory into the original concept of Stackelberg solutions. It is shown that Stackelberg problems under fuzziness are transformed into deterministic bilevel linear or nonlinear programming problems, and that the resulting problems are exactly solved by using conventional bilevel linear or nonlinear programming techniques.

In this paper the author analyses generalization of the classical quadratic programming problem, when along with linear constraints quadratic constraints are allowed. The author consider the case when variables can take only binary values 0 or 1. Such problems are important when choice of the optimal structures of systems covers a large number of variants, and the choice or refusal to select individual variants is equivalent to values 1 or 0 of the corresponding binary variables. There is described the procedure of reducing the indicated problem to the fourth-degree polynomial programming problem on the basis of the penalty method, when all terms are no greater than in the second power, except one having a fourth power. The solution of the obtained optimization problem is reduced to solving a system of equations containing only binary variables. The author proposes a heuristic recursive algorithm for solving the resulting system of equations, and variants of constructing the initial version of the solution. The parameters used in the proposed method are described. In the process of reduction, there are used the classical Cardano formulas for the roots of a cubic equation, for the practical finding of which the relations are obtained and the calculation procedure is described. The algorithm given in this paper can also be used to solve many problems of discrete mathematics.

A parallel algorithm has been proposed for solving the problem of construction of nonlinear models (mathematical expressions, functions, algorithms, and programs) using given experimental data, set of variables, basic functions and operations. The designed algorithm of multivariant evolutionary synthesis of nonlinear models includes linear representation of a chromosome, modular operations in decoding of a genotype into a phenotype for interpreting a chromosome as a sequence of instructions, and a multi-variant method for presenting a set of models (expressions) using a single chromosome. A sequential version of the algorithm is compared with a standard genetic programming (GP) algorithm and a Cartesian genetic programming (CGP) one. The algorithm proposed was shown to excel the GP and CGP algorithms both in the time required for search for a solution (more than by an order of magnitude in most cases) and in the probability of finding a given function (model). Experiments have been carried out on parallel supercomputer systems, and estimates of the efficiency of the parallel algorithm offered have been obtained; the estimates demonstrate linear acceleration and scalability.

In order to recover a signal from its compressive measurements, the compressed sensing theory seeks the sparsest signal that agrees with the measurements, which is actually an l norm minimization problem. In this paper, we equivalently transform the l norm minimization into a concave continuous piecewise linear programming, and propose an optimization algorithm based on a modified interior point method. Numerical experiments demonstrate that our algorithm improves the sufficient number of measurements, relaxes the restrictions of the sensing matrix to some extent, and performs robustly in the noisy scenarios.

This paper works on a modified simplex algorithm for the local optimization of Continuous PieceWise Linear (CPWL) programming with generalization of hinging hyperplane objective and linear constraints. CPWL programming is popular since it can be equivalently transformed into difference of convex functions programming or concave optimization. Inspired by the concavity of the concave CPWL functions, we propose an Objective Variation Simplex Algorithm (OVSA), which is able to find a local optimum in a reasonable time. Computational results are presented for further insights into the performance of the OVSA compared with two other algorithms on random test problems.

In this paper, we investigate the computational behavior of the exterior point simplex algorithm. Up until now, there has been a major difference observed between the theoretical worst case complexity and practical performance of simplex-type algorithms. Computational tests have been carried out on randomly generated sparse linear problems and on a small set of benchmark problems. Specifically, 6780 linear problems were randomly generated, in order to formulate a respectable amount of experiments. Our study consists of the measurement of the number of iterations that the exterior point simplex algorithm needs for the solution of the above mentioned problems and benchmark dataset. Our purpose is to formulate representative regression models for these measurements, which would play a significant role for the evaluation of an algorithm’s efficiency. For this examination, specific characteristics, such as the number of constraints and variables, the sparsity and bit length, and the condition of matrix A, of each linear problem, were taken into account. What drew our attention was that the formulated model for the randomly generated problems reveal a linear relation among these characteristics.

To globally solve linear multiplicative programming problem (LMP), this paper presents a practicable branch-and-bound method based on the framework of branch-and-bound algorithm. In this method, a new linear relaxation technique is proposed firstly. Then, the branch-and-bound algorithm is developed for solving problem LMP. The proposed algorithm is proven that it is convergent to the global minimum by means of the subsequent solutions of a series of linear programming problems. Some experiments are reported to show the feasibility and efficiency of this algorithm.

Most of the current methods for solving linear fractional programming (LFP) problems depend on the simplex type method. In this paper, we present a new approach for solving linear fractional programming problem in which the objective function is a linear fractional function, while constraint functions are in the form of linear inequalities. This approach does not depend on the simplex type method. Here first we transform this LFP problem into linear programming (LP) problem and hence solve this problem algebraically using the concept of duality. Two simple examples to illustrate our algorithm are given. And also we compare this approach with other available methods for solving LFP problems.

An algorithm is proposed in this paper for solving two-dimensional bi-level linear programming problems without making a graph. Based on the classification of constraints, algorithm removes all redundant constraints, which eliminate the possibility of cycling and the solution of the problem is reached in a finite number of steps. Example to illustrate the method is also included in the paper.

In this paper we attempt to develop a parametric simplex algorithm for solving biobjective convex separable piecewise linear programming problems. The algorithm presented in this paper can be regarded as an extension of the parametric simplex algorithm for solving biobjective linear programming problems to the piecewise linear case. Analogous to the linear case, this parametric simplex algorithm provides a decomposition of parametric space. A numerical example is presented to illustrate the implementation of the algorithm.

Posterior constraint optimal selection techniques (COSTs) are developed for nonnegative linear programming problems (NNLPs), and a geometric interpretation is provided. The posterior approach is used in both a dynamic and non-dynamic active-set framework. The computational performance of these methods is compared with the CPLEX standard linear programming algorithms, with two most-violated constraint approaches, and with previously developed COST algorithms for large-scale problems.

This paper deals with developing an efficient algorithm for solving the fully fuzzy linear fractional programming problem. To this end, we construct a new method which is obtained from combination of Charnes−Cooper scheme and the multi-objective linear programming problem. Furthermore, the application of the proposed method in real life problems is presented and this method is compared with some existing methods. The numerical experiments and comparative results presented promising results to find the fuzzy optimal solution.

We present a direct algorithm for solving the vertical generalized linear complementarity problem, first considered by Cottle and Dantzig, when the associated matrix is a vertical block P-matrix. The algorithm converges to a unique solution in a finite number of steps, without an assumption of nondegeneracy on the given problem. The algorithm is simple, efficient, and easy to implement.

We consider a class of nonlinear integer optimization problems commonly known as fractional 0---1 programming problems (also, often referred to as hyperbolic 0---1 programming problems), where the objective is to optimize the sum of ratios of affine functions subject to a set of linear constraints. Such problems arise in diverse applications across different fields, and have been the subject of study in a number of papers during the past few decades. In this survey we overview the literature on fractional 0---1 programs including their applications, related computational complexity issues and solution methods including exact, approximation and heuristic algorithms.

The computational complexity of linear and nonlinear programming problems depends on the number of objective functions and constraints involved and solving a large problem often becomes a difficult task. Redundancy detection and elimination provides a suitable tool for reducing this complexity and simplifying a linear or nonlinear programming problem while maintaining the essential properties of the original system. Although a large number of redundancy detection methods have been proposed to simplify linear and nonlinear stochastic programming problems, very little research has been developed for fuzzy stochastic (FS) fractional programming problems. We propose an algorithm that allows to simultaneously detect both redundant objective function(s) and redundant constraint(s) in FS multi-objective linear fractional programming problems. More precisely, our algorithm reduces the number of linear fuzzy fractional objective functions by transforming them in probabilistic-possibilistic constraints characterized by predetermined confidence levels. We present two numerical examples to demonstrate the applicability of the proposed algorithm and exhibit its efficacy.

© 2016 Elsevier Inc. Multilevel (including bi-level and tri-level) programming aims to solve decentralized decision-making problems that feature interactive decision entities distributed throughout a hierarchical organization. Since the multilevel programming problem is strongly NP-hard and traditional exact algorithmic approaches lack efficiency, heuristics-based particle swarm optimization (PSO) algorithms have been used to generate an alternative for solving such problems. However, the existing PSO algorithms are limited to solving linear or small-scale bi-level programming problems. This paper first develops a novel bi-level PSO algorithm to solve general bi-level programs involving nonlinear and large-scale problems. It then proposes a tri-level PSO algorithm for handling tri-level programming problems that are more challenging than bi-level programs and have not been well solved by existing algorithms. For the sake of exploring the algorithms' performance, the proposed bi/tri-level PSO algorithms are applied to solve 62 benchmark problems and 810 large-scale problems which are randomly constructed. The computational results and comparison with other algorithms clearly illustrate the effectiveness of the proposed PSO algorithms in solving bi-level and tri-level programming problems.

This paper presents the convergence proof and complexity analysis of an interior-point framework that solves linear programming problems by dynamically selecting and adding relevant inequalities. First, we formulate a new primal–dual interior-point algorithm for solving linear programmes in non-standard form with equality and inequality constraints. The algorithm uses a primal–dual path-following predictor–corrector short-step interior-point method that starts with a reduced problem without any inequalities and selectively adds a given inequality only if it becomes active on the way to optimality. Second, we prove convergence of this algorithm to an optimal solution at which all inequalities are satisfied regardless of whether they have been added by the algorithm or not. We thus provide a theoretical foundation for similar schemes already used in practice. We also establish conditions under which the complexity of such algorithm is polynomial in the problem dimension and address remaining limitations without these conditions for possible further research.

A second-order learning algorithm based on differential geometry is used to numerically solve the linear matrix equation Q = x + ∑ i = 1 m A i T x A i − ∑ i = 1 n B i T x B i . An extended Hamiltonian algorithm is proposed based on the manifold of symmetric positive definite matrices. The algorithm is compared with traditional coupled fixed-point algorithm. Numerical experiments illustrate that the convergence speed of the provided algorithm is faster than that of the coupled fixed-point algorithm.

We develop an efficient homogeneous and self-dual interior-point method (IPM) for the linear programs arising in economic model predictive control of constrained linear systems with linear objective functions. The algorithm is based on a Riccati iteration procedure, which is adapted to the linear system of equations solved in homogeneous and self-dual IPMs. Fast convergence is further achieved using a warm-start strategy. We implement the algorithm in MATLAB and C. Its performance is tested using a conceptual power management case study. Closed loop simulations show that: 1) the proposed algorithm is significantly faster than several state-of-the-art IPMs based on sparse linear algebra and 2) warm-start reduces the average number of iterations by 35%–40%.

An algorithm for computing the covering constant for the restriction of a linear operator to a cone defined by a finite set of inequalities is proposed. After a finite number of steps, the algorithm reduces the original problem to one of finding the eigenvalues of linear operators.

Linear Programming (LP) finds many uses in business and industry, where a decision maker may want to utilize limited available resources in the best possible manner. Limited resources may include material, money, manpower, space and time. Linear programming provides various methods of solving such problems. This paper presents a new approach for selecting constraints in solving a linear programming problem. Here the intercept values are used to find out the equality constraints among the inequality constraints. This new method will be valid in achieving optimal solution.

In this paper, a new method namely, objective separable method based on Linear Programming with Bounded Variables Algorithm is proposed for finding an optimal solution to a Quasi-Concave Quadratic Programming Problems with Bounded Variables in which the objective function involves the product of two indefinite factorized linear functions and the constraint functions are in the form of linear inequalities. For developing this method, we use programming language MATHEMATICA. We also illustrate numerical examples to demonstrate our method.Dhaka Univ. J. Sci. 64(1): 51-58, 2016 (January)

Generalized linear multiplicative programming problems are difficult problems known to be N P -hard, and they frequently arise in various areas such as engineering, economics, finance, etc. At present, most of the global optimization algorithms for this kind of problems are mainly developed based on those kinds of branch and bound algorithms. In this study, a new algorithm based on the concept of level set is presented for globally solving generalized linear multiplicative programming problems. The proposed algorithm is concise and easy to implement, so it is quite suitable for engineering applications. Finally, the global convergence is presented and the practical efficiency of the algorithm is demonstrated by the numerical experiments.

A numerical algorithm for minimizing a convex function on a smooth surface is proposed. The algorithm is based on reducing the original problem to a sequence of convex programming problems. Necessary extremum conditions are examined, and the convergence of the algorithm is analyzed.

In this paper, we propose a new hybrid social spider algorithm with simplex Nelder-Mead method in order to solve integer programming and minimax problems. We call the proposed algorithm a Simplex Social Spider optimization (SSSO) algorithm. In the the proposed SSSO algorithm, we combine the social spider algorithm with its powerful capability of performing exploration, exploitation, and the Nelder-Mead method in order to refine the best obtained solution from the standard social spider algorithm. In order to investigate the general performance of the proposed SSSO algorithm, we test it on 7 integer programming problems and 10 minimax problems and compare against 10 algorithms for solving integer programming problems and 9 algorithms for solving minimax problems. The experiments results show the efficiency of the proposed algorithm and its ability to solve integer and minimax optimization problems in reasonable time.

A projection algorithm for positive definite linear complementarity problems is introduced. The algorithm incorporates the basic idea of gradient projection algorithms. A new descent direction, which will be referred to as the vector of centers, is used instead of the negative of the objective function gradient. This descent direction is kept fixed throughout the iterations, thus reducing the computational effort of computing a search direction by per step. The algorithm exploits the ellipsoid algorithm to find an advanced starting point on the boundary of the feasible region. Accordingly, the algorithm is warm-started and the number of iterations required to reach the solution is reduced. Starting from this point, the algorithm generates a series of boundary points that approaches the solution by projecting the proposed descent direction onto the feasible region. A numerical example is solved and a practical example in the simulation of rigid body dynamics is given to demonstrate the usage of the...

We propose a polynomial algorithm for a separable convex optimization problem with linear constraints. We do not make any additional assumptions about the structure of the objective function except for polynomial computability. That is, the objective function can be non-differentiable. The running time of our algorithm is polynomial in the size of the input consisting of an instance of the problem and the accuracy with which the problem is to be solved. Our algorithm uses an oracle for solving auxiliary systems of linear inequalities. This oracle can be any polynomial algorithm for linear programming.

We construct an algorithm for the numerical solution of nonlinear third-order boundary value problems. This algorithm is based on eight-point binary subdivision scheme. Proposed algorithm is stable and convergent and gives more accurate results than fourth-degree B-spline algorithm.

The paper presents a technique for solving the binary linear programming model in polynomial time. The general binary linear programming problem is transformed into a convex quadratic programming problem. The convex quadratic programming problem is then solved by interior point algorithms. This settles one of the open problems of whether P = NP or not. The worst case complexity of interior point algorithms for the convex quadratic problem is polynomial. It can also be shown that every liner integer problem can be converted into binary linear problem.

In this paper, we have suggested a new technique to transform multi-objective linear programming problem (MOLPP) to the single objective linear programming problem by using Harmonic mean for values of function and an algorithm is suggested for its solution, the computer application of algorithm has been demonstrated by solving some numerical examples. We have used some other techniques, such as (sen, arithmetic mean, median) to solve the same problems, the results in Table 3 indicate that the new technique in general is promising.