Tropical optimization problems in time-constrained project scheduling

We consider a project that consists of activities to be performed in parallel under various temporal constraints, which include start-start, start-finish and finish-start precedence relationships, release times, deadlines and due dates. Scheduling problems are formulated to find optimal schedules for the project with respect to different objective functions to be minimized, such as the project makespan, the maximum deviation from the due dates, the maximum flow-time and the maximum deviation of finish times. We represent these problems as optimization problems in terms of tropical mathematics, and then solve them by applying direct solution methods of tropical optimization. As a result, new direct solutions of the scheduling problems are obtained in a compact vector form, which is ready for further analysis and practical implementation. The solutions are illustrated by simple numerical examples.     

Reduction of the bilevel stochastic optimization problem with quantile objective function to a mixed‐integer problem

The paper is devoted to the stochastic optimistic bilevel optimization problem with quantile criterion in the upper level problem. If the probability distribution is finite, the problem can be transformed into a mixed‐integer nonlinear optimization problem. We formulate assumptions guaranteeing that an optimal solution exists. A production planning problem is used to illustrate usefulness of the model. Copyright © 2017 John Wiley & Sons, Ltd.     

On optimizing a linear objective function subjected to fuzzy relation inequalities

In this paper, we extend Guo and Xia’s necessary condition which has been presented by Guo and Xia (Fuzzy optimizat Decis Mak 5: 33–47, 2006) in order to study the finitely many constraints of fuzzy relation inequalities and optimize a linear objective function on this region which is defined by the fuzzy max–min operator. The new condition provides a means for removing the unnecessary paths resulting from Guo and Xia’s paths. Also, an algorithm and two numerical examples are offered to abbreviate and illustrate the steps of the resolution process of the problem.     

线性约束三次规划问题的全局最优性必要条件和最优化算法

讨论了带线性不等式约束三次规划问题的最优性条件和最优化算法. 首先, 讨论了带有线性不等式约束三次规划问题的 全局最优性必要条件. 然后, 利用全局最优性必要条件, 设计了解线性约束三次规划问题的一个新的局部最优化算法(强局部最优化算法). 再利用辅助函数和所给出的新的局部最优化算法, 设计了带有线性不等式约束三 规划问题的全局最优化算法. 最后, 数值算例说明给出的最优化算法是可行的、有效的.     

A transformation technique for optimal control problems with partially linear state inequality constraints

This paper considers optimal control problems involving the minimization of a functional subject to differential constraints, terminal constraints, and a state inequality constraint. The state inequality constraint is of a special type, namely, it is linear in some or all of the components of the state vector.A transformation technique is introduced, by means of which the inequality-constrained problem is converted into an equality-constrained problem involving differential constraints, terminal constraints, and a control equality constraint. The transformation technique takes advantage of the partial linearity of the state inequality constraint so as to yield a transformed problem characterized by a new state vector of minimal size. This concept is important computationally, in that the computer time per iteration increases with the square of the dimension of the state vector.In order to illustrate the advantages of the new transformation technique, several numerical examples are solved by means of the sequential gradient-restoration algorithm for optimal control problems involving nondifferential constraints. The examples show the substantial savings in computer time for convergence, which are associated with the new transformation technique.This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-76-3075, and by the National Science Foundation, Grant No. MCS-76-21657.     

Two new algorithms for solving optimization problems with one linear objective function and finitely many constraints of fuzzy relation inequalities

This paper studies the optimization model of a linear objective function subject to a system of fuzzy relation inequalities (FRI) with the max-Einstein composition operator. If its feasible domain is non-empty, then we show that its feasible solution set is completely determined by a maximum solution and a finite number of minimal solutions. Also, an efficient algorithm is proposed to solve the model based on the structure of FRI path, the concept of partial solution, and the branch-and-bound approach. The algorithm finds an optimal solution of the model without explicitly generating all the minimal solutions. Some sufficient conditions are given that under them, some of the optimal components of the model are directly determined. Some procedures are presented to reduce the search domain of an optimal solution of the original problem based on the conditions. Then the reduced domain is decomposed (if possible) into several sub-domains with smaller dimensions that finding the components of the optimal solution in each sub-domain is very easy. In order to obtain an optimal solution of the original problem, we propose another more efficient algorithm which combines the first algorithm, these procedures, and the decomposition method. Furthermore, sufficient conditions are suggested that under them, the problem has a unique optimal solution. Also, a comparison between the recently proposed algorithm and the known ones will be made.     

A single machine scheduling problem with availability constraints and sequence-dependent setup costs

We study a single machine scheduling problem with availability constraints and sequence-dependent setup costs, with the aim of minimizing the makespan. To the authors’ knowledge, this problem has not been treated as such in the operations research literature. We derive in this paper a mixed integer programming model to deal with such scheduling problem. Computational tests showed that commercial solvers are capable of solving only small instances of the problem. Therefore, we propose two ways for reducing the execution time, namely a valid inequality that strengthen the linear relaxation and an efficient heuristic procedure that provides a starting feasible solution to the solver. A substantial gain is achieved both in terms of the linear programming relaxation bound and in terms of the time to obtain an integer optimum when we use the enhanced model in conjunction with providing to the solver the solution obtained by the proposed heuristic.     

A unified framework for linear control problems with state-variable inequality constraints

This paper briefly reviews the literature on necessary optimality conditions for optimal control problems with state-variable inequality constraints. Then, it attempts to unify the treatment of linear optimal control problems with state-variable inequality constraints in the framework of continuous linear programming. The duality theory in this framework makes it possible to relate the adjoint variables arising in different formulations of a problem; these relationships are illustrated by the use of a simple example. This framework also allows more general problems and admits a simplex-like algorithm to solve these problems.This research was partially supported by Grant No. A4619 from the National Research Council of Canada to the first author. The first author also acknowledges the support provided by the Brookhaven National Laboratory, where he conducted his research.     

Analytical solutions to the optimization of a quadratic cost function subject to linear and quadratic equality constraints

In the area of broad-band antenna array signal processing, the global minimum of a quadratic equality constrained quadratic cost minimization problem is often required. The problem posed is usually characterized by a large optimization space (around 50–90 tuples), a large number of linear equality constraints, and a few quadratic equality constraints each having very low rank quadratic constraint matrices. Two main difficulties arise in this class of problem. Firstly, the feasibility region is nonconvex and multiple local minima abound. This makes conventional numerical search techniques unattractive as they are unable to locate the global optimum consistently (unless a finite search area is specified). Secondly, the large optimization space makes the use of decision-method algorithms for the theory of the reals unattractive. This is because these algorithms involve the solution of the roots of univariate polynomials of order to the square of the optimization space. In this paper we present a new algorithm which exploits the structure of the constraints to reduce the optimization space to a more manageable size. The new algorithm relies on linear-algebra concepts, basic optimization theory, and a multivariate polynomial root-solving tool often used by decision-method algorithms.This research was supported by the Australian Research Council and the Corporative Research Centre for Broadband Telecommunications and Networking.     

Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints

In this paper, we propose an easily implementable algorithm in Hilbert spaces for solving some classical monotone variational inequality problem over the set of solutions of mixed variational inequalities. The proposed method combines two strategies: projected subgradient techniques and viscosity-type approximations. The involved stepsizes are controlled and a strong convergence theorem is established under very classical assumptions. Our algorithm can be applied for instance to some mathematical programs with complementarity constraints.     

A full-Newton step infeasible interior-point method for linear optimization based on a trigonometric kernel function

An infeasible interior-point method (IIPM) for solving linear optimization problems based on a kernel function with trigonometric barrier term is analysed. In each iteration, the algorithm involves a feasibility step and several centring steps. The centring step is based on classical Newton’s direction, while we used a kernel function with trigonometric barrier term in the algorithm to induce the feasibility step. The complexity result coincides with the best-known iteration bound for IIPMs. To our knowledge, this is the first full-Newton step IIPM based on a kernel function with trigonometric barrier term.     

On the calculation of a membership function for the solution of a fuzzy linear optimization problem

In the present paper the fuzzy linear optimization problem (with fuzzy coefficients in the objective function) is considered. Recent concepts of fuzzy solution to the fuzzy optimization problem based on the level-cut and the set of Pareto optimal solutions of a multiobjective optimization problem are applied. Chanas and Kuchta suggested one approach to determine the membership function values of fuzzy optimal solutions of the fuzzy optimization problem, which is based on calculating the sum of lengths of certain intervals. The purpose of this paper is to determine a method for realizing this idea. We derive explicit formulas for the bounds of these intervals in the case of triangular fuzzy numbers and show that only one interval needs to be considered.     

Duality for multiobjective optimization problems with convex objective functions and D.C. constraints

In this paper we provide a duality theory for multiobjective optimization problems with convex objective functions and finitely many D.C. constraints. In order to do this, we study first the duality for a scalar convex optimization problem with inequality constraints defined by extended real-valued convex functions. For a family of multiobjective problems associated to the initial one we determine then, by means of the scalar duality results, their multiobjective dual problems. Finally, we consider as a special case the duality for the convex multiobjective optimization problem with convex constraints.     

A random coordinate descent algorithm for optimization problems with composite objective function and linear coupled constraints

In this paper we propose a variant of the random coordinate descent method for solving linearly constrained convex optimization problems with composite objective functions. If the smooth part of the objective function has Lipschitz continuous gradient, then we prove that our method obtains an ?-optimal solution in $\mathcal{O}(n^{2}/\epsilon)$ iterations, where n is the number of blocks. For the class of problems with cheap coordinate derivatives we show that the new method is faster than methods based on full-gradient information. Analysis for the rate of convergence in probability is also provided. For strongly convex functions our method converges linearly. Extensive numerical tests confirm that on very large problems, our method is much more numerically efficient than methods based on full gradient information.     

Random problem generation and the computation of efficient extreme points in multiple objective linear programming

This paper looks at the task of computing efficient extreme points in multiple objective linear programming. Vector maximization software is reviewed and the ADBASE solver for computing all efficient extreme points of a multiple objective linear program is described. To create MOLP test problems, models for random problem generation are discussed. In the computational part of the paper, the numbers of efficient extreme points possessed by MOLPs (including multiple objective transportation problems) of different sizes are reported. In addition, the way the utility values of the efficient extreme points might be distributed over the efficient set for different types of utility functions is investigated. Not surprisingly, results show that it should be easier to find good near-optimal solutions with linear utility functions than with, for instance, Tchebycheff types of utility functions.Dedicated to Professor George B. Dantzig on the occasion of his eightieth birthday.     

The effect of strengthened linear formulations on improving the lower bounds for the part families with precedence constraints problem

The part families with precedence constraints problem (PFP) arises in industry, when flexible manufacturing systems are designed within a group technology approach. The aim of this problem is to arrange parts into families by imposing capacity constraints, concerning both the number of parts and processing times, besides precedence constraints in the building of families.     

A robust optimization approach to diet problem with overall glycemic load as objective function

The main purpose of the paper is to introduce a mixed-integer programming model for the diet problem with glycemic load (GL) values of foods as objective function parameters. It is assumed that the glycemic load values are subject to uncertainty. The diet problem with minimum cost function is well-known in the literature. However, the diet problem with minimum total daily GL values of foods that satisfies the daily nutritional and serving size requirements has not been proposed. Robust optimization approach is used to account for uncertainty in the GL values of foods. The decision maker is flexible to tune the degree of uncertainty rather than assuming a worst-case scenario. An experimental analysis with a total of 177 foods is performed based on the nutritional and serving size requirements and the basic food groups recommended by the U.S. Department of Health and Human Services & U.S. Department of Agriculture (USDA). The results of the experimental analysis with different scenarios give different solutions for different degrees of uncertainty. However, some foods are frequently found to be in the optimum solutions. These foods are in good agreement with the literature advising them as a part of a daily diet for attaining low level of blood glucose levels. Although we believe that the proposed diet problem with minimum total GL has contributions for satisfying the daily nutritional and serving size requirements with a minimum level of effect on blood glucose levels, it has several limitations. It is a basic diet problem, and assumes that the overall GL is a linear combination of number of serving sizes with the GL values of foods. It also does not consider any other factors such as several combinations of foods and their varying effects on blood glucose levels. These factors should be considered for the next research.     

A GENERALIZED PROJECTION-SUCCESSIVE LINEAREQUATIONS ALGORITHM FOR NONLINEARLY EQUALITy AND INEQUALITY CONSTRAINED OPTIMIZATION AND ITS RATE OF CONVERGENCE

In this paper, a new superlinearly convergent algorithm is presented for optimization problems with general nonlineer equality and inequality Constraints, Comparing with other methods for these problems, the algorithm has two main advantages. First, it doesn‘t solve anyquadratic programming (QP), and its search directions are determined by the generalized projection technique and the solutions of two systems of linear equations. Second, the sequential points generated by the algoritbh satisfy all inequity constraints and its step-length is computed by the straight line search,The algorithm is proved to possesa global and auperlinear convergence.     

Multi-criteria assignment problem with incompatibility and capacity constraints

The considered assignment problem generalizes its classical counterpart by the existence of some incompatibility constraints limiting the assignment of tasks to processing units within groups of mutually exclusive tasks. The groups are defined for each processing unit and the constraints allow at most one task from each group to be assigned to the corresponding processing unit. The processing units can normally process a certain number of tasks without any cost; this capacity can be extended, however, at some extra marginal cost that is non-decreasing with the number of additional tasks. Each task has to be assigned to exactly one processing unit and has some preference for the assignment; it is expressed for each pair ‘task-processing unit’ by a dissatisfaction degree. The quality of feasible assignments is evaluated by three criteria: g ₁-the maximum dissatisfaction of tasks, g ₂-the total dissatisfaction of tasks, g ₃-the total cost of processing units. If there is no feasible assignment, tasks and processing units creating a blocking configuration are identified and all actions of unblocking are proposed. Formal properties of blocking configurations and unblocking actions are proven, and an interactive procedure for exploring the set of non-dominated assignments is described together with illustrative examples processed by special software.     

Distributed computing viewed as a team problem with constraints

The optimal distribution of the workload in a system of interconnected computer units is considered. Formulated as a team decision problem with a singular cost criterion and with equality and inequality constraints, it is shown that the problem admits always a unique piecewise linear strategy which is globally optimal. Some interesting particular cases are studied.The research reported in this paper was made possible through support from the Office of Naval Research under the Joint Services Electronics Program by Contract No. N00014-75-C-0648 and Contract No. N00014-77-C-0531 and by the National Science Foundation, Grant No. ENG-76-11824.