Class Of Optimization Problems Research Articles

The Probabilistic Method for Solving Minimum Vertex Cover Problem Using Systems of Non-linear Equations

In this paper the probabilistic method is presented for solving the minimum vertex cover problem using systems of non-linear equations that are formed on the basis of a neighborhood relationship of a particular vertex of a given graph. The minimum vertex cover problem is one of the classic mathematical optimization problems that have been shown to be NP-hard. It has a lot of real-world applications in different fields of science and technology. This study finds solutions to this problem by means of the two basic procedures. In the first procedure three probabilistic pairs of variables according to the maximum vertex degree are formed and processed accordingly. The second procedure checks a given graph for the presence of the leaf vertices. Special software package to check the validity of these procedures was written. The experiment results show that our method has significantly better time complexity and much smaller frequency of the approximation errors in comparison with one of the most currently efficient algorithms.

Optimal Control Vector Parameterization Approach with a Hybrid Intelligent Algorithm for Nonlinear Chemical Dynamic Optimization Problems

AbstractA control vector parameterization (CVP)‐based hybrid algorithm, HAPSODSA‐CVP, is proposed to solve the nonlinear chemical dynamic optimization problems, where adaptive particle swarm optimization (APSO) is applied to enhance the global search ability, while differential search algorithm (DSA) is used to improve the local exploitation ability. Three well‐known classic nonlinear chemical dynamic optimization problems are tested as illustration, and detailed comparisons are carried out among PSO‐CVP, APSO‐CVP, and HAPSODSA‐CVP approaches. The research results not only demonstrate the efficiency of the HAPSODSA‐CVP approach for this kind of dynamic optimization problems, but also its superiority to both APSO‐CVP and PSO‐CVP approaches in terms of accuracy as well as convergence rate.

Improving the Performance of Metaheuristics: An Approach Combining Response Surface Methodology and Racing Algorithms

The setup of heuristics and metaheuristics, that is, the fine-tuning of their parameters, exercises a great influence in both the solution process, and in the quality of results of optimization problems. The search for the best fit of these algorithms is an important task and a major research challenge in the field of metaheuristics. The fine-tuning process requires a robust statistical approach, in order to aid in the process understanding and also in the effective settings, as well as an efficient algorithm which can summarize the search process. This paper aims to present an approach combining design of experiments (DOE) techniques and racing algorithms to improve the performance of different algorithms to solve classical optimization problems. The results comparison considering the default metaheuristics and ones using the settings suggested by the fine-tuning procedure will be presented. Broadly, the statistical results suggest that the fine-tuning process improves the quality of solutions for different instances of the studied problems. Therefore, by means of this study it can be concluded that the use of DOE techniques combined with racing algorithms may be a promising and powerful tool to assist in the investigation, and in the fine-tuning of different algorithms. However, additional studies must be conducted to verify the effectiveness of the proposed methodology.

The robust (minmax regret) single machine scheduling with interval processing times and total weighted completion time objective

Single machine scheduling is a classical optimization problem that depicts multiple real life systems in which a single resource (the machine) represents the whole system or the bottleneck operation of the system. In this paper we consider the problem under a weighted completion time performance metric in which the processing time of the tasks to perform (the jobs) are uncertain, but can only take values from closed intervals. The objective is then to find a solution that minimizes the maximum absolute regret for any possible realization of the processing times. We present an exact branch-and-bound method to solve the problem, and conduct a computational experiment to ascertain the possibilities and limitations of the proposed method. The results show that the algorithm is able to optimally solve instances of moderate size (25–40 jobs depending on the characteristics of the instance).

最优分红策略:正则与脉冲混合控制问题

In this paper, we revisit the classic optimal dividend problem in a diffusion model. Note that, the aggregated dividend payments process is a cadlag process, whose path can be partitioned into two parts: Continuous part and discontinuous part. Therefore, we incorporate both the continuous dividend form and the impulse dividend form in our model, which leads to a regular-impulse combined stochastic control problem. We assume that there exist a maximum rate for continuous dividend payments and both proportional and fixed costs for impulse dividend payments. When the company's surplus is described by a drifted Brownian motion, and the objective is to maximize the expectation of total discounted dividend payments up to the time of ruin, we obtain the explicit closed-form expressions for the value function and the optimal dividend policy. The optimal dividend policy is shown to be a combinational form of the threshold policy and the impulse policy.

Optimization in first semester calculus: a look at a classic problem

Optimization problems in first semester calculus have historically been a challenge for students. Focusing on the classic optimization problem of finding the minimum amount of fencing required to enclose a fixed area, we examine students' activity through the lens of Tall and Vinner's concept image and Carlson and Bloom's multidimensional problem-solving framework. We are particularly interested in students' evoked concept images for the mathematical concepts that play a role in the construction of the function whose minimum is needed. Data analysis revealed several gaps in students’ understanding of the optimization process. We highlight the mathematical concepts that play a role in these gaps, focusing on variables, function notation, function composition, properties of rectangles and the relationships between them, the role of the optimizing function, and the graphical representation of the function.

An iterative multi-objective particle swarm optimization-based control vector parameterization for state constrained chemical and biochemical engineering problems

Dynamic optimization of the state constrained chemical and biochemical engineering problems has always been the research hotspot due to the difficulties of handling constraints. In this paper, a novel approach, CVP–IMOPSO, is presented to tackle this kind of problems, where the original optimization problem is firstly converted into a multi-objective dynamic optimization problem based on a method of handling state constraint; and control vector parameterization (CVP) is then applied to transform the resulting infinite dimensional problem into a nonlinear programming (NLP) problem; finally, an efficient iterative multi-objective particle swarm optimization (IMOPSO), embedded with an region reduction strategy, is proposed to tackle this MOO problem. This strategy reduces the search space gradually during the iteration process so as to further promote the convergence rate and diversity of MOPSO. Three well-known classic optimization problems for chemical and biochemical engineering processes have been tested as illustration, and the detailed comparisons among IMOPSO, MOPSO and NSGA-II are carried out for seven benchmark problems. The research results not only show that the proposed IMOPSO is an efficient algorithm for MOO problems, but also reveal the effectiveness of the proposed approach.

Solving non-linear portfolio optimization problems with interval analysis

Estimation errors or uncertainities in expected return and risk measures create difficulties for portfolio optimization. The literature deals with the uncertainty using stochastic, fuzzy or probability programming. This paper proposes a new approach to treating uncertainty. By assuming that the expected return and risk vary within a bounded interval, this paper uses interval analysis to extend the classical mean-variance portfolio optimization problem to the cases with bounded uncertainty. To solve the interval quadratic programming problem, the paper adopts order relations to transform the uncertain programme into a deterministic programme, and includes the investors’ risk preference into the model. Numerical analysis illustrates the advantage of this new approach against conventional methods.

Spatial resource allocation via extremal optimization enhanced by cell-based local search

A new treatment is presented for land use planning problems by means of extremal optimization (EO) in conjunction to cell-based neighborhood local search. EO, inspired by self-organized critical models of evolution has been applied mainly to the solution of classical combinatorial optimization problems. Cell-based local search has been employed by the author elsewhere in problems of spatial resource allocation in combination with genetic algorithms and simulated annealing. In this paper, it complements EO in order to enhance its capacity for a spatial optimization problem. The hybrid method thus formed is compared to methods of the literature on a specific resource allocation problem by taking into account both the development and the transportation cost. It yields better results both in terms of objective function values and in terms of compactness. The latter is an important quantity for spatial planning and its meaning is discussed. The appearance of significant compactness values as emergent results is investigated.

Disorder problem for a Brownian motion on a segment in the case of uniformly distributed moment of disorder

The paper deals with the quickest detection of the disorder of a Brownian motion on a finite interval. The unknown moment of disorder is assumed to be uniformly distributed on the interval. The Bayesian and absolute criteria are used as optimality tests. The problem is reduced to a classic optimal stopping problem where the optimal stopping time may be obtained as a solution to integral equations. The existence and uniqueness of the solution of integral equations are proved analytically.

Design optimization based on state problem functionals

This paper presents a general mathematical structure for design optimization problems, where state problem functionals are used as design objectives.It extends to design optimization the general model of physical theories pioneered by Tonti (1972, 1976) and Oden and Reddy (1974, 1983). It turns out that the classical structural optimization problem of compliance minimization is a member of the treated general class of problems. Other particular examples, discussed in the paper, are related to Darcy-Stokes flow and pipe flow models. A main novel feature of the paper is the unification of seemingly different design problems, but the general mathematical structure also explains some previously not fully understood phenomena. For instance, the self-penalization property of Stokes flow design optimization receives an explanation in terms of minimization of a concave function over a convex set.

Vertex Cover Meets Scheduling

We consider a hybrid two-stage optimization problem that generalizes two classic combinatorial optimization problems: (i) weighted vertex cover in graphs, and (ii) makespan minimization in multiprocessor scheduling. An instance specifies a machine environment, a set of jobs, and an undirected graph over the jobs. The goal is to select a subset of the jobs that forms a vertex cover and to schedule it on a set of parallel machines so as to minimize the makespan. We call this problem family vertex cover meets multiprocessor scheduling (VCMS). The problem is motivated by networks where vertices represent servers and each edge corresponds to a task that can be done by any of the two servers of its endpoint. Each selected server can complete any number of tasks assigned to it within a given time defined by its weight, as the time consumption of the server is roughly equal to its activation time. The activation is performed by a set of $$m$$m processors, such that every selected server is activated by one processor, and the goal is to minimize the maximum total activation time assigned to any processor. We design a multitude of approximation algorithms for VCMS and its variants, many of which match or almost match the best approximation bound known for the vertex cover problem. In particular, we give a $$2$$2-approximation for the case of a fixed number of unrelated machines, a $$3$$3-approximation for an arbitrary number of unrelated machines, and a $$(2+\varepsilon )$$(2+?)-approximation for an arbitrary number of identical machines. Furthermore we consider special graph classes for which the weighted vertex cover problem can be solved to optimality in polynomial time: for many of these classes, there is a PTAS for VCMS on identical machines; for bipartite graphs, however, VCMS on identical machines turns out to be APX-hard. Finally, we study the bin packing counterpart of VCMS and design a $$(2+\varepsilon )$$(2+?)-approximation algorithm for it.

Generalized interval vector spaces and interval optimization

This paper presents a method for endowing the generalized interval space with some different structures, such as vector spaces, order relations and an algebraic calculus. With these concepts we formulate interval optimization problems and relate them to classic multi-objective optimization problems. We also present a version of the Von Neumann’s Mini-max Theorem in the interval context.

Nested optimization method combining complex method and ant colony optimization to solve JSSP with complex associated processes

Job Shop Scheduling Problem (JSSP) is one of classic combinatorial optimization problems and has a long research history. Modern job shop has following characteristics: increasingly complicated processes, small batch and personalized requirement, which lead to complex correlations among processes. Complex correlations of processes, involving nested correlations besides serial and parallel correlations, propose a new task for JSSP research. Decomposing JSSP into two nested sub problems of order of arranging processes and machine arrangement, this research integrates the traditional thought of complex method into the ant colony optimization (ACO) to develop a nested optimization method in order to solve the new task. This paper is divided into four parts: first, the model of JSSP with complex associated processes is constructed and the difficulties to solve which are analyzed and listed; second, the definition of “order of arranging processes” is originally proposed, based on which the mathematical model available for the complex method is developed, taking process starting time as design variables of the first level optimization. The steps of the first level optimization and the secondary nested flow chart are detailed with the demonstration of the effectiveness of the complex method’s iteration mechanism; third, based on the representation of features the order of arranging processes obtained by the first level optimization combined with the first-in first-out rule owns, the corresponding modified ACO algorithm, involving pheromone positive perception and reverse spreading mechanism, is put forward to realize the second level optimization, which result is taken as the objective function value of the complex vertex to realize the secondary nested optimization strategy; finally, taking plentiful JSSP with complex associated processes as study cases, a serial of comparative experiments are done respectively adopting the genetic algorithm, ACO algorithm, particle swarm optimization algorithm, some combinations of heuristic algorithms respectively in the nested two levels, and the proposed nested optimization method, and experiment results attest the reliability and superiority of the proposed method.

An assembly sequence planning approach with a rule-based multi-state gravitational search algorithm

Assembly sequence planning (ASP) becomes one of the major challenges in product design and manufacturing. A good assembly sequence leads to reduced costs and duration in the manufacturing process. However, assembly sequence planning is known to be a classical NP-hard combinatorial optimization problem; ASP with many product components becomes more difficult to solve. In this paper, an approach based on a new variant of the gravitational search algorithm (GSA) called the rule-based multi-state gravitational search algorithm (RBMSGSA) is used to solve the assembly sequence planning problem. As in the gravitational search algorithm, the RBMSGSA incorporates Newton’s law of gravity, the law of motion, and a rule that makes each assembly component of each individual solution occur once based on precedence constraints; the best feasible sequence of assembly can then be determined. To verify the feasibility and performance of the proposed approach, a case study has been performed and a comparison has been conducted against other three approaches based on simulated annealing (SA), a genetic algorithm (GA), and binary particle swarm optimization (BPSO). The experimental results show that the proposed approach has achieved significant improvement in performance over the other methods studied.

A multi-criteria methodology for optimising road investments

There is growing awareness that road investment appraisal, at both project and network levels, requires joint consideration of multiple criteria that relate to engineering, economic, social and environmental impacts. Several multi-criteria analytical methodologies exist for use at the project level. At the network level, strategies and works programmes need to be optimised under budget constraints by considering the set of relevant criteria. This paper presents a multi-criteria analysis methodology that can be used to optimise road network investments using the utility/cost ratio method. The multi-criteria analysis methodology is based upon the analytical hierarchy process technique, which systematically transforms the analysis of competing objectives into a series of simple comparisons between the constituent elements. The multi-criteria analysis methodology developed was applied to generate a work programme using a set of selected criteria to which relative weights of preference were assigned. The results obtained were compared with those of a similar work programme generated using conventional economic analysis in order to highlight the issue of effectiveness in road management and the use of multi-criteria analysis to solve a classical multi-objective optimisation problem under budget constraints.

Three-Dimensional Path Planning Method for Autonomous Underwater Vehicle Based on Modified Firefly Algorithm

Path planning is a classic optimization problem which can be solved by many optimization algorithms. The complexity of three-dimensional (3D) path planning for autonomous underwater vehicles (AUVs) requires the optimization algorithm to have a quick convergence speed. This work provides a new 3D path planning method for AUV using a modified firefly algorithm. In order to solve the problem of slow convergence of the basic firefly algorithm, an improved method was proposed. In the modified firefly algorithm, the parameters of the algorithm and the random movement steps can be adjusted according to the operating process. At the same time, an autonomous flight strategy is introduced to avoid instances of invalid flight. An excluding operator was used to improve the effect of obstacle avoidance, and a contracting operator was used to enhance the convergence speed and the smoothness of the path. The performance of the modified firefly algorithm and the effectiveness of the 3D path planning method were proved through a varied set of experiments.

Análisis computacional de los problemas del vendedor viajero y patrones de corte

The aim of this article is to perform a computational study to analyze the impact of formulations, and the solution strategy on the algorithmic performance of two classical optimization problems: the traveling salesman problem and the cutting stock problem. In order to assess the algorithmic performance on both problems three dependent variables were used: solution quality, computing time and number of iterations. The results are useful for choosing the solution approach to each specific problem. In the STSP, the results demonstrate that the multistage decision formulation is better than the conventional formulations, by solving 90.47% of the instances compared with MTZ (76.19%) and DFJ (14.28%). The results of the CSP demonstrate that the cutting patterns formulation is better than the standard formulation with symmetry breaking inequalities, when the objective function is to minimize the loss of trim when cutting the rolls.

An Approximate Algorithm for Triangle TSP with a Four-Vertex-Three-Line Inequality

Traveling salesman problem (TSP) is a classic combinatorial optimization problem. The time complexity of the exact algorithms is generally an exponential function of the scale of TSP. This work gives an approximate algorithm with a four-vertex-three-line inequality for the triangle TSP. The time complexity is O(n2) and it can generate an approximation less than 2 times of the optimal solution. The paper designs a simple algorithm with the inequality. The algorithm is compared with the double-nearest neighbor algorithm. The experimental results illustrate the algorithm find the better approximations than the double-nearest neighbor algorithm for most TSP instances.

A Convex Analytic Approach to Risk-Aware Markov Decision Processes

In classical Markov decision process (MDP) theory, we search for a policy that, say, minimizes the expected infinite horizon discounted cost. Expectation is, of course, a risk neutral measure, which does not suffice in many applications, particularly in finance. We replace the expectation with a general risk functional, and call such models risk-aware MDP models. We consider minimization of such risk functionals in two cases, the expected utility framework, and conditional value-at-risk, a popular coherent risk measure. Later, we consider risk-aware MDPs wherein the risk is expressed in the constraints. This includes stochastic dominance constraints, and the classical chance-constrained optimization problems. In each case, we develop a convex analytic approach to solve such risk-aware MDPs. In most cases, we show that the problem can be formulated as an infinite-dimensional linear program (LP) in occupation measures when we augment the state space. We provide a discretization method and finite approximations for solving the resulting LPs. A striking result is that the chance-constrained MDP problem can be posed as an LP via the convex analytic method.