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We are interested in time evolution of systems that switch their modes of operation at discrete moments of time. The intervals between switching may, in general, vary. The number of modes may be finite or infinite. The mathematical setting for such systems is variable time step dynamics with choice. We have used this setting previously to study the long term behavior of such systems. In this paper, we define and study the continuous time dynamics whose trajectories are limits of trajectories of discrete systems as time step goes to zero. The limit dynamics is multivalued. In the special case of a switched system, when the dynamics is generated by switching between solutions of a finite number of systems of ODEs, we show that our continuous limit solution set coincides with the solution set of the relaxed differential inclusion.

Reusable components and libraries reduce costs in software development but also bring challenges like ensuring that application's components form a consistent and working set. While dependency management and build tools provide assistance in creating the set, they can't guarantee its correctness in terms of interoperability. On the other hand, the methods which detect component interoperability issues do not provide guidance in finding the proper set of components to fix any uncovered inconsistencies. In this work we present a method for finding such set of components which provides the required functionality, is free from type-level inconsistencies, and at the same time is optimal according to a given criterion. The method is based on pre-computed compatibility data and integer linear programming and allows to optimize the found solution set with respect to an arbitrary cost function.

Cellular layout includes inter-cell and intra-cell layouts. This paper presented a multi-objective-integrated optimization model that minimizes the total material handling distance and the area of the cellular layout. To avoid the decrease of the solution set of cellular layout problem by solving the inter-cell and intra-cell layouts separately, the proposed model solves the two steps simultaneously. To handle the complexity confronted with the large-scale model, a structured real-code cooperative particle swarm optimization (CPSO) was proposed, the particles were divided into genic particles and non-genic particles, operations of genetic algorithm (GA) were applied on the two kinds of particles, and the performance of the particle swarm optimization was improved largely. Simulation experiment verified the validity of the model and algorithm.

Levenberg-Marquardt (L-M forshort) method is one of the most important methods for solving systems of nonlinear equations. In this paper, we consider the convergence under \(\lambda_{k}=\min(\|F_{k}\|,\|J_{k}^{T}F_{k}\|)\) of L-M method. We will show that if ∥ F(x k ) ∥ provides a local error bound, which is weaker than the condition of nonsingularity for the system of nonlinear equations, the sequence generated by the L-M method converges to the point of the solution set quadratically. As well, numerical experiments are reported.

We propose an iterative improvement method for an enclosure of the solution set of a system of interval linear equations. The method sequentially cuts off (shaves) parts of a given enclosure that contain no solution, yielding thus tighter enclosures. Since shaving can be done independently in the coordinates, the procedure is easily parallelized. Our approach is convenient for problems with wide input intervals, where traditional methods give poor enclosures. Finally, we present a limited computational study.

In this work we present a new simple but efficient scheme – Subsquares approach – for development of algorithms for enclosing the solution set of overdetermined interval linear systems. We are going to show two algorithms based on this scheme and discuss their features. We start with a simple algorithm as a motivation, then we continue with an improved algorithm. Both algorithms can be easily parallelized. The features of both algorithms will be discussed and numerically tested.

This paper proposes a Population-P-Systems-inspired Membrane Algorithm (PPSMA) for multi-objective optimization. In the algorithm, the cells of population P systems are divided into two groups to implement different functions and the communications among cells are performed at two levels in order to obtain well converged and distributed solution set. Moreover, differential evolution is employed as search operator in PPSMA. Twelve multi-objective benchmark problems are utilized to test algorithm performance. Experimental results show that PPSMA performs better than five compared algorithms.

We can derive several inclusion and interference results between the minimum sets of quadratic functions and the pre-kernel. These results enable us to give a full characterization of the pre-kernel in terms of constrained minimum sets, or restricted sub-differential of the corresponding conjugations of quadratic functions, that is, we implicitly base the representation of the pre-kernel on the Fenchel-Moreau conjugation of the characteristic function. In a further step additional results related to the vector spaces of balanced excesses are attained which allow us to give a replication result. Having worked out these auxiliary results we then turn our attention to the issue whether it is possible to replicate any arbitrary payoff vector on the domain as a pre-kernel element of a game constructed from a payoff equivalence class that contains this payoff vector. There, we provide an impossibility theorem. Moreover, we also address the reverse issue if any pre-kernel solution of a default game can be supported as a pre-kernel element of a related game from the same game space. This issue can be partly affirmed. It is shown that any pre-kernel belonging to a payoff set which satisfies the non-empty interior property is replicable as a pre-kernel element of a related game. From the replication result further results on the structure of the pre-kernel are established, for instance, on its disconnectedness.

In many applications, it is required to optimize several conflicting objectives concurrently leading to a multobjective optimization problem (MOP). The solution set of a MOP, the Pareto set, typically forms a (k-1)-dimensional object, where k is the number of objectives involved in the optimization problem. The purpose of this chapter is to give an overview of recently developed set oriented techniques - subdivision and continuation methods - for the computation of Pareto sets \(\mathcal{P}\) of a givenMOP. All these methods have in common that they create sequences of box collections which aim for a tight covering of \(\mathcal{P}\). Further, we present a class of multiobjective optimal control problems which can be efficiently handled by the set oriented continuation methods using a transformation into high-dimensionalMOPs. We illustrate all the methods on both academic and real world examples.

This paper intends to propose an energy functional based modeling technique on an n-phase multiphase motor. In motor control area, the multiphase motor is becoming more and more popular recently. The multiphase can be applied in direct-drive electric vehicle. However, the associated mathematical model for energy functional is seldom discussed. This paper will discuss the modeling of the motor system by energy functional optimization. Orthogonal particle swarm optimization (OPSO) is used to derive the optimal solution set for the dynamic system. The Simulation and experimental results shows the validity of the proposed model. It is believed that the developed system model can be used in the energy functional of the multiphase motor.

The problem of computing the hull, that is the tightest interval enclosure of the solution set for linear systems with parameters being nonlinear functions of interval parameters, is an NP-hard problem. However, since the problem of computing the hull can be considered as a combinatorial or as a constrained optimisation problem, metaheuristic techniques might be helpful. Alas, experiments performed so far show that they are time consuming and their performance may depend on the problem size and structure, therefore some acceleration and stabilisation techniques are required. In this paper, a new approach which rely on a multi-agent system is proposed. The idea is to apply evolutionary method and differential evolution for different agents working together to solve constrained optimisation problems. The results obtained for several examples from structural mechanics involving many parameters with large uncertainty ranges show that some synergy effect of the metaheuristics can be achieved, especially for problems of a larger size.

A direction based multi-objective agent genetic algorithm DMOAGA is proposed in this paper. In order to take advantage of the effective direction information and depth of local search to mine non-dominated solutions, the direction perturbation operator is also employed. The neighborhood non-dominated solutions are generated using tournament selection and "average distance" rule, which maintains the diversity of non-dominated solution set. In the experiments, the benchmark problems UF1~UF6 and ZDT1~ZDT4 are used to validate the performance of DMOAGA. We compared it with NSGA-II and DMEA in terms of generational distance GD and inverted generational distance IGD. The results show that DMOAGA has a good diversity and convergence, the performances on most of benchmark problems are better than DMEA and NSGA-II.

Many industrial optimization cases present themselves in a multi-objective (MO) setting (where each of the objectives portrays different aspects of the problem). Therefore, it is important for the decision-maker to have a solution set of options prior to selecting the best solution. In this work, the weighted sum scalarization approach is used in conjunction with three meta-heuristic algorithms; differential evolution (DE), chaotic differential evolution (CDE) and gravitational search algorithm (GSA). These methods are then used to generate the approximate Pareto frontier to the green sand mould system problem. The Hypervolume Indicator (HVI) is applied to gauge the capabilities of each algorithm in approximating the Pareto frontier. Some comparative studies were then carried out with the algorithms developed in this work and that from the previous work. Analysis on the performance as well as the quality of the solutions obtained by these algorithms is shown here.

In this note a concept of e-level set function is introduced, i.e. a function which approximates a level set function satisfying the Hamilton-Jacobi inequality. We prove that each Lipschitz continuous solution of the Hamilton-Jacobi inequality is an e-level set function. Next, a numerical approximation of the level set function is presented, i.e. method for the construction of an e-level set function.

The polyhedral cone feasibility problem PCFP that occupied us in the last two chapters, though fundamental, is better understood when regarded within the more general context of linear programming. Succinctly described, the latter is a family of problems that consist in optimizing (i.e., maximizing or minimizing) a linear function over a set defined by linear constraints (equalities and/or inequalities).

An optimization model with a 1inear objective function subject to a system of fuzzy relation equations was presented. Since the non-empty feasible solution set of the fuzzy relation equations was generally a non-convex set, a particle swarm optimization (PSO) algorithm was proposed. The PSO algorithm didn’t more complex by the rising of degree of the problem, and could avoid being trapped in local optimum due to using alterable inertia weight. We applied the proposed algorithm to an example, and compared its result with those generated by GA algorithms. The experimental comparison demonstrates that the performance of PSO algorithm is competitive with others and will be an effective method.

The problem of solving linear systems whose coefficients are nonlinear functions of parameters varying within prescribed intervals is investigated. A new method for outer interval solution of such system is proposed. In order to reduce memory usage, nonlinear dependencies between parameters are handled using revised affine arithmetic. Some numerical experiments which aim to show the properties of the proposed method are reported.

We present a new method for solving multivalued variational inequalities, where the underlying function is upper semicontinuous and satisfies a certain generalized monotone assumption. First, we construct an appropriate hyperplane which separates the current iterative point from the solution set. Then the next iterate is obtained as the projection of the current iterate onto the intersection of the feasible set with the halfspace containing the solution set. We also analyze the global convergence of the algorithm under minimal assumptions.

Yard allocation is the key point of container terminal management. This paper solves the problem by minimizing the distance between the berth and the yard, offering multiple work ways, avoiding loading or collecting at the same time. The solution of yard allocation is similar with the model which all individual organisms have the same possibility to be killed in Unified Neutral Theory of Biodiversity and Biogeography. Based on the improvement of the neutral theory, this paper proposed a multiple group model to reduce the solution set. To guarantee the adjacent container group not to be put in the same yard and assure a consecutive and parallel loading process, a strategy which kills anyone of the species exist prey relationship is put forward. And a greedy strategy is proposed to select an island to accelerate the ecological balance among islands. All these strategies make ecological selection more instructive and faster to find an optimal solution. The experiments show that the model proposed in this paper owns a good performance.

In group decision making (GDM) framework, we focus on decision problems defined under uncertainty where decision makers can hesitate among several values to elicit their preferences. In such cases, the use of hesitant fuzzy linguistic term sets (HFLTS) can facilitate the elicitation of decision makers preferences. In this contribution, our aim is to propose a linguistic GDM model that allows to decision makers use single linguistic terms or comparative linguistic terms to express their preferences and obtain the solution set of alternatives of the GDM problem.

In this paper, we propose an interactive fuzzy satisficing method for multiobjective stochastic linear programming problems to obtain a satisficing solution, in which the criteria of probability maximization and fractile optimization are considered simultaneously. In the proposed method, it is assumed that the decision maker has fuzzy goals for not only permissible objective levels of a probability maximization model but also permissible probability levels of a fractile criterion optimization model. After eliciting the corresponding membership functions for the fuzzy goals, two kinds of membership functions for permissible objective levels and permissible probability levels are integrated through the fuzzy decision. An interactive algorithm is proposed to obtain a satisficing solution from among a D f -Pareto optimal solution set.

We show that several variants of the problem k-Dominating Set, including k-Connected Dominating Set, k-Independent Dominating Set, k-Dominating Clique, d-Distancek-Dominating Set, k-Perfect Code and d-Distancek-Perfect Code, when parameterized by the solution size k, remain W[1]-hard in either multiple-interval graphs or their complements or both.

In this chapter we collect some results in Nonlinear Analysis that will be fre- quently used in the book. The first part of this chapter deals with comparison prin- ciples for second order differential operators and enables us to obtain an ordered structure of the solution set and, in most of the cases, the uniqueness of the solution. In the second part of this chapter we review the celebrated method of moving planes that allows us to deduce the radial symmetry of the solution. The third part of this chapter is concerned with variational methods. The final section contains some re- sults in degree theory that will be mostly used to derive existence and nonexistence of a stationary solution to some reaction-diffusion systems.

The Viewing Graph [1] represents several views linked by the corresponding fundamental matrices, estimated pairwise. Given a Viewing Graph, the tuples of consistent camera matrices form a family that we call the Solution Set. This paper provides a theoretical framework that formalizes different properties of the topology, linear solvability and number of solutions of multi-camera systems. We systematically characterize the topology of the Viewing Graph in terms of its solution set by means of the associated algebraic bilinear system. Based on this characterization, we provide conditions about the linearity and the number of solutions and define an inductively constructible set of topologies which admit a unique linear solution. Camera matrices can thus be retrieved efficiently and large viewing graphs can be handled in a recursive fashion. The results apply to problems such as the projective reconstruction from multiple views or the calibration of camera networks.

Previously we presented a new approach to solving dynamic constraint satisfaction problems (DCSPs) based on detection of major bottlenecks in a problem using a weighted-degree method called "random probing". The present work extends this approach and the analysis of the performance of this algorithm. We first show that despite a reduction in search effort, variability in search effort with random probing after problem perturbation is still pronounced, reflected in low correlations between performance measures on the original and perturbed problems. Using an analysis that separates effects based on promise and fail-firstness, we show that such variability is mostly due to variation in promise. Moreover, the stability of fail-firstness is greater when random probiing is used than with non-adaptive heuristics. We then present an enhancement of our original probing procedure, called "random probing with solution guidance", which improves average performance (as well as solution stability). Finally, we present an analysis of the nearest solution in the perturbed problem to the solution found for the original (base) problem. These results show why solution repair methods do poorly when problems are in a critical complexity region, since there may be no solutions similar to the original one in the perturbed problem. They also show that on average probing with solution guidance finds solutions with near-maximal stability under these conditions.

This paper was targeted at minimizing the expectation of traveling distance maximizing the expectation of customers' degree satisfaction, a multi-objective vehicle routing problem with stochastic demand (VRPSD) model based on soft time window was proposed. In order to solve the problem, a hybrid PSO algorithm based on Pareto optimization method was designed in this paper. The paper made the standard PSO algorithm discrete by re-defining operators and employing swap recon, utilized challenge tournament method to construct Pareto optimal solution set, applied an external archive to keep the diversity of solutions. Ultimately, a standard example is used to verify the validity of the algorithm.

In this contribution our aim is to present a multicriteria linguistic decision making model in which experts might provide their assessments by using linguistic expressions based on comparative terms close to the expressions used by human beings in real world problems or single linguistic terms. To aggregate such a type of linguistic information two symbolic aggregation operators are introduced. Finally, an exploitation phase is proposed to build a preference relation among alternatives and then, a non-dominance choice degree is applied to obtain the solution set of alternatives.

Theoretical economics makes use of strict mathematical methods. For instance, games as introduced by von Neumann and Morgenstern allow for formal mathematical proofs for certain axiomatized economical situations. Such proofs can--at least in principle--also be carried through in formal systems such as Theorema. In this paper we describe experiments carried through using the Theorema system to prove theorems about a particular form of games called pillage games. Each pillage game formalizes a particular understanding of power. Analysis then attempts to derive the properties of solution sets (in particular, the core and stable set), asking about existence, uniqueness and characterization. Concretely we use Theorema to show properties previously proved on paper by two of the co-authors for pillage games with three agents. Of particular interest is some pseudo-code which summarizes the results previously shown. Since the computation involves infinite sets the pseudocode is in several ways non-computational. However, in the presence of appropriate lemmas, the pseudo-code has sufficient computational content that Theorema can compute stable sets (which are always finite). We have concretely demonstrated this for three different important power functions.

The multiobjective fuzzy genetic clustering technique described in the previous chapter simultaneously optimizes the Xie-Beni (XB) index [442] and the fuzzy Cmeans (FCM) [62] measure (Jm). In multiobjective optimization (MOO), a search is performed over a number of, often conflicting, objective functions. Instead of yielding a single best solution, MOO yields the final solution set containing a number of non-dominated Pareto-optimal solutions. A characteristic of the multiobjective fuzzy clustering approach is that it often produces a large number of Pareto-optimal solutions, from which selecting a particular solution is difficult.

Study the particle swarm optimization algorithm for solving multi-objective optimization problems. This paper presents a set select space of multi-objective particle swarm optimization algorithm (SMOPSO) based on the Pareto dominate relations, using the method of fast sort to construct non-dominated solution set, pre-set an upper limit of non-dominated solution space, and set an external set to save the optimal solutions. At the same time introduce genetic algorithm crossover and mutation operator for each generation of the updated part of the particles in order to increase the diversity of particle groups, so we can get good distribution of particles. The simulation results verify the validity of this method.

Pathway analysis is a powerful tool to study metabolic reaction networks under steady state conditions. An Elementary pathway constitutes a minimal set of reactions that can operate at steady state such that each reaction also proceeds in the appropriate direction. In mathematical terms, elementary pathways are the extreme rays of a polyhedral cone--the solution set of homogeneous equality and inequality constraints. Enumerating all extreme rays--given the constraints--is difficult especially if the problem is degenerate and high dimensional. We present and compare two approaches for the parallel enumeration of extreme rays, both based on the double description method. This iterative algorithm has proven efficient especially for degenerate problems, but is difficult to parallelize due to its sequential operation. The first approach parallelizes single iteration steps individually. In the second approach, we introduce a born/die matrix to store intermediary results, allowing for parallelization across several iteration steps. We test our multicore implementations on a 16 core machine using large examples from combinatorics and biology.

The construction and operation of modern power station units on coal base are subject to considerable economic, ecological and political constraints enforcing the system’s elements to operate at their technical upper limits. The control and prevention of fuel-caused harmful effects is crucial for profits and losses with respect to availability, efficiency and maintenance expenses. Advanced power plant technologies are an indefensible high-risk without adequate developed monitoring and optimization systems. A real-time optimization method of the flame position in a multi-burner firing system that is affected by corrosion processes at the evaporative heating surfaces is presented. The base is a model of the flame position generated by artificial neural networks. Evolutionary algorithms are used to optimize the fuel distribution which depends on the feeding of the coal dust burners. By analyzing the current plant status and technical restrictions for the respective components the solution set is reduced to a single solution which is realized as control action.

A challenging multi-objective sequencing problem with variable launching intervals has been studied. A novel hybrid algorithm based on a multi-objective clonal selection algorithm and a co-evolutionary algorithm has been developed for the system control. The clonal selection algorithm for the multi-objective sequencing models is worked as a driving system, while the co-evolutionary immune algorithm for acquiring launching intervals is subordinated and run in parallel on distributed systems in order to guarantee the real-time requirements. The evolution operators such as coding, decoding and collaboration formation mechanism are defined. The scheme has been proven to improve the system optimization and achieve better solution sets as compared with other available algorithms.

An interval global optimization method combined with the Direct Method for solving parametric linear systems is used for computing a tight enclosure for the solution set of parametric linear system whose input data are non-linear functions of interval parameters. Revised affine arithmetic is used to handle the nonlinear dependencies. The Direct Method performs the monotonicity test to speed up the convergence of the global optimization. It is shown that the monotonicity test significantly increases the convergence of the global optimization method. Some illustrative examples are solved by the discussed method, and the results are compared to literature data produces by other methods.

In this paper, we elaborate how decision space diversity can be integrated into indicator-based multiobjective search. We introduce DIOP, the diversity integrating multiobjective optimizer, which concurrently optimizes two set-based diversity measures, one in decision space and the other in objective space. We introduce a possibility to improve the diversity of a solution set, where the minimum proximity of these solutions to the Pareto-front is user-defined. Experiments show that DIOP is able to optimize both diversity measures and that the decision space diversity can indeed be improved if the required maximum distance of the solutions to the front is relaxed.

In this paper, a new approach for solving first order fuzzy differential equations (FDEs) with fuzzy initial value is considered under strongly generalized H-differentiability. In order to obtain solution of FDE, we extend the 1-cut solution of original problem. This extension is constructed based on the allocating some unknown spreads to 1-cut solution, then created value is replaced in the original FDE. However obtaining solutions of FDE is equivalent to determine the unknown spreads while 1-cut solution is derived via previous step (in general, 1-cut of FDE is interval differential equation). Moreover, we will introduce three new set solutions for FDEs based on the concepts of united solution set, tolerable solution set and controllable solution set. Indeed, our approach is designed to obtain such new solution sets while one of them has pessimistic/optimitic attitude. Finally, some numerical examples are solved to illustrate the approach.

This paper puts forward a brief runtime analysis of an evolutionary programming (EP) which is one of the most important continuous optimization evolutionary algorithms. A theoretical framework of runtime analysis is proposed by modeling EP as an absorbing Markov process. The framework is used to study the runtime of a classical EP algorithm named as EP with Cauchy mutation (FEP). It is proved that the runtime of FEP can be less than a polynomial of n if the Lebesgue measure of optimal solution set is more than an exponential form of 2. Moreover, the runtime analysis result can be used to explain the performance of EP based on Cauchy mutation.

For the problem of fuzzy comprehensive evaluation, use the method of Tsukamoto to find the weight of solution sets of fuzzy evaluation in inverse problems; normalize the multiple weight solutions that given by the actual conditions, then optimize it by the method of lattice close-degree, and get the optimal weight solution; and use this method to do decision-making for enterprise management system. The example that the fuzzy evaluation method of weight optimization is applied into the business management decision making proves the method is reasonable and effective.

In this article we present a new approach to teaching computer science - the evaluation and visual modeling of algorithms based on metaphorical forms - applied within the core of a virtual education system, the development module for computer-based lessons (LDS). We reveal the structure and characteristics of the teaching process that we implemented in the proposed system, students and their roles, applied teaching methods, solutions for evaluation and a case study on a lesson model. We presented the state of the art in this domain highlighting the advantages of the described solution set, and also possible extensions.

Conventional set approximations are based on a set of attributes; however, these approximations cannot relate an object to the corresponding attribute. In this study, a new model for set approximation based on individual attributes is proposed for interval-valued data. Defining an indiscernibility relation is omitted since each attribute value itself has a set of values. Two types of approximations, single- and multiattribute approximations, are presented. A multi-attribute approximation has two solutions: a maximum and a minimum solution. A maximum solution is a set of objects that satisfy the condition of approximation for at least one attribute. A minimum solution is a set of objects that satisfy the condition for all attributes. The proposed set approximation is helpful in finding the features of objects relating to condition attributes when interval-valued data are given. The proposed model contributes to feature extraction in interval-valued information systems.

The paper presents a new class of adaptive and sequentially guaranteeing PPS-methods, based on partitioning parameter sets, for computing optimal (exact) component-wise bounds of the solution sets to interval linear systems with square regular matrices.

As pointed out in the previous chapter, it is imperative that the MOEA must be capable of attaining high convergence speeds in order to find the optimal solution set before it changes and becomes obsolete. However, high convergence speed often implies a rapid loss of diversity during the optimization process, which inevitably leads to the inability to track the dynamic Pareto front. Therefore, it is necessary to maintain or generate sufficient diversity to explore the search space when the multi-objective problem changes.