Bound Sets Research Articles

ULTIMATE BOUND FOR AN ACTIVE LORENZ SYSTEM

Considering the Lorenz-like model [1] derived from Active Model H for scalar active matter introduced by Tiribocchi et al. [2], we investigate the ultimate bound and global attractive sets for the system, and obtain the ultimate bound characterized by an ellipsoid and a global exponential attractive set for the system.

Hamilton energy, competitive modes and ultimate bound estimation of a new 3D chaotic system, and its application in chaos synchronization

Abstract This paper discusses the dynamical behavior of a new 3D chaotic system of integer and fractional order. To get a comprehensive knowledge of the dynamics of the proposed system, we have studied competitive modes and Hamilton energy for different parameter values. In order to get the ultimate bound set for the proposed system, we employed the Lagrange coefficient approach to solve the optimization problem. We have also explored the use of the bound set in synchronization. Furthermore, we have examined the Hamilton energy, time series, bifurcation diagrams, and Lyapunov exponents for the fractional version of the proposed chaotic system. Finally, we looked at the Mittage-Leffler positive invariant sets and global attractive sets by merging the Lyapunov function approach with the Mittage-Leffler function. Numerical simulations have shown the obtained bound sets and other analytical outcomes.

Advancements in the computation of enclosures for multi-objective optimization problems

A central goal for multi-objective optimization problems is to compute their nondominated sets. In most cases these sets consist of infinitely many points and it is not a practical approach to compute them exactly. One solution to overcome this problem is to compute an enclosure, a special kind of coverage, of the nondominated set. For that computation one often makes use of so-called local upper bounds. In this paper we present a generalization of this concept. For the first time, this allows to apply a warm start strategy to the computation of an enclosure. We also show how this generalized concept allows to remove empty areas of an enclosure by deleting certain parts of the lower and upper bound sets which has not been possible in the past. We demonstrate how to apply our ideas to the box approximation algorithm, a general framework to compute an enclosure, as recently used in the solver called BAMOP. We show how that framework can be simplified and improved significantly, especially concerning its practical numerical use. In fact, we show for selected numerical instances that our new approach is up to eight times faster than the original one. Hence, our new framework is not only of theoretical but also of practical use, for instance for continuous convex or mixed-integer quadratic optimization problems.

An algorithmic approach to multiobjective optimization with decision uncertainty

In real life applications, optimization problems with more than one objective function are often of interest. Next to handling multiple objective functions, another challenge is to deal with uncertainties concerning the realization of the decision variables. One approach to handle these uncertainties is to consider the objectives as set-valued functions. Hence, the image of one decision variable is a whole set, which includes all possible outcomes of this decision variable. We choose a robust approach and thus these sets have to be compared using the so-called upper-type less order relation. We propose a numerical method to calculate a covering of the set of optimal solutions of such an uncertain multiobjective optimization problem. We use a branch-and-bound approach and lower and upper bound sets for being able to compare the arising sets. The calculation of these lower and upper bound sets uses techniques known from global optimization, as convex underestimators, as well as techniques used in convex multiobjective optimization as outer approximation techniques. We also give first numerical results for this algorithm.

Column generation algorithms for bi-objective combinatorial optimization problems with a min–max objective

Many practical combinatorial optimization problems can be described by integer linear programs having an exponential number of variables, and they are efficiently solved by column generation algorithms. For these problems, column generation is used to compute good dual bounds that can be incorporated in branch-and-price algorithms. Recent research has concentrated on describing lower and upper bounds of bi-objective and general multi-objective problems with sets of points (bound sets). An important issue to address when computing a bound set by column generation is how to efficiently search for columns corresponding to each point of the bound set. In this work, we propose a generalized column generation scheme to compute bound sets for bi-objective combinatorial optimization problems. We present specific implementations of the generalized scheme for the case where one objective is a min–max function by using a variant of the $$\varepsilon $$ -constraint method to efficiently model these problems. The proposed strategies are applied to a bi-objective extension of the multi-vehicle covering tour problem, and their relative performances based on different criteria are compared. The results show that good bound sets can be obtained in reasonable times if columns are efficiently managed. The variant of the $$\varepsilon $$ -constraint presented is also better than a standard $$\varepsilon $$ -constraint method in terms of the quality of the bound sets.

Surrogate upper bound sets for bi-objective bi-dimensional binary knapsack problems

The paper deals with the definition and the computation of surrogate upper bound sets for the bi-objective bi-dimensional binary knapsack problem. It introduces the Optimal Convex Surrogate Upper Bound set, which is the tightest possible definition based on the convex relaxation of the surrogate relaxation. Two exact algorithms are proposed: an enumerative algorithm and its improved version. This second algorithm results from an accurate analysis of the surrogate multipliers and the dominance relations between bound sets. Based on the improved exact algorithm, an approximated version is derived. The proposed algorithms are benchmarked using a dataset composed of three groups of numerical instances. The performances are assessed thanks to a comparative analysis where exact algorithms are compared between them, the approximated algorithm is confronted to an algorithm introduced in a recent research work.

History of ESA's Historical Records Committee

History of ESA's Historical Records Committee

Sonar Image Enhancement based on Minimum Entropy of One-Dimensional Bound Histogram and Rough Sets

To enhance a sonar image from a small underwater target, the method using the minimum entropy of the one-dimensional bound histogram and Rough Set Theory is proposed. In the method, the threshold of the gray values of the sonar image is calculated via the minimum value of the entropy of the one-dimensional bound histogram from the sonar image, and the sonar image is divided into some varied sub-images via Rough Set Theory. The varied sub-images are enhanced via their corresponding transforms in gray scale of the sonar image. As an example, an actual sonar image from a small underwater target is enhanced via the proposed method with detailed steps. As a contrast, the sonar image is also enhanced via transformation of histogram equalization. The experimental results show the proposed method is more effective in both improving visualization of the sonar image and preserving its edges.

Algorithmic improvements on dynamic programming for the bi-objective {0,1} knapsack problem

This paper presents several methodological and algorithmic improvements over a state-of-the-art dynamic programming algorithm for solving the bi-objective {0,1} knapsack problem. The variants proposed make use of new definitions of lower and upper bounds, which allow a large number of states to be discarded. The computation of these bounds are based on the application of dichotomic search, definition of new bound sets, and bi-objective simplex algorithms to solve the relaxed problem. Although these new techniques are not of a common application for dynamic programming, we show that the best variants tested in this work can lead to an average improvement of 10 to 30 % in CPU-time and significant less memory usage than the original approach in a wide benchmark set of instances, even for the most difficult ones in the literature.

Solving the multidimensional knapsack problems with generalized upper bound constraints by the adaptive memory projection method

An adaptive memory projection (referred as AMP) method is developed for multidimensional knapsack problems (referred as the MKP) with generalized upper bound constraints. All the variables are divided into several generalized upper bound (referred as GUB) sets and at most one variable can be chosen from each of the GUB sets. The MKP with GUBs (referred as the GUBMKP) can be applied to many real-world problems, such as capital budgeting, resource allocation, cargo loading, and project selection. Due to the complexity of the GUBMKP, good metaheuristics are sought to tackle this problem. The AMP method keeps track of components of good solutions during the search and creates provisional solution by combining components of better solutions. The projection method, which can free the selected variables while fixing the others, is very useful for metaheuristics, especially when tackling large-scale combinatorial optimization. In this paper, the AMP method is implemented by iteratively using critical event tabu search as a search routine, and CPLEX in the referent optimization stage. Variables that are strongly determined, consistent, or attractive, are identified in the search process. Selected variables from this pool are fed into CPLEX as a small subproblem. In addition to the diversification effect within critical event tabu search, the pseudo-cut inequalities and an adjusted frequency penalty scalar are also applied to increase opportunities of exploring new regions. This study conducts a comprehensive sensitivity analysis on the parameters and strategies used in the proposed AMP method. The computational results show several variants of the AMP method outperforms the tight oscillation method in the literature of GUBMKP. On average, consistent variables tend to perform best as a pure strategy. A pure strategy equipped with local search can lead into even better results. Last but not least, testing different types of variables in the referent optimization stage before selecting just one of the pure strategies is found to be very helpful.

A two phase method for multi-objective integer programming and its application to the assignment problem with three objectives

In this paper, we present a generalization of the two phase method to solve multi-objective integer programmes with p > 2 objectives. We apply the method to the assignment problem with three objectives. We have recently proposed an algorithm for the first phase, computing all supported efficient solutions. The second phase consists in the definition and the exploration of the search area inside of which nonsupported nondominated points may exist. This search area is not defined by trivial geometric constructions in the multi-objective case, and is therefore difficult to describe and to explore. The lower and upper bound sets introduced by Ehrgott and Gandibleux in 2001 are used as a basis for this description. Experimental results on the three-objective assignment problem where we use a ranking algorithm to explore the search area show the efficiency of the method.

Bound sets for biobjective combinatorial optimization problems

In this paper we introduce the concept of bound sets for multiobjective discrete optimization. We prove general results on lower and upper bound sets for combinatorial optimization problems with multiple objectives. We present general algorithms for constructing lower and upper bound sets for biobjective problems and provide numerical results on five different problem types.

The multiscenario lot size problem with concave costs

The dynamic single-facility single-item lot size problem is addressed. The finite planning horizon is divided into several time periods. Although the total demand is assumed to be a fixed value, the distribution of this demand among the different periods is unknown. Therefore, for each period the demand can be chosen from a discrete set of values. For this reason, all the combinations of the demand vector yield a set of different scenarios. Moreover, we assume that the production/reorder and holding cost vectors can vary from one scenario to another. For each scenario, we consider as the objective function the sum of the production/reorder and the holding costs. The problem consists of determining all the Pareto-optimal or non-dominated production plans with respect to all scenarios. We propose a solution method based on a multiobjective branch and bound approach. Depending on whether shortages are considered or not, different upper bound sets are provided. Computational results on several randomly generated problems are reported.

Information-theoretical approach to control of quantum-mechanical systems

Fundamental limits on the controllability of quantum mechanical systems are discussed in the light of quantum information theory. It is shown that the amount of entropy-reduction that can be extracted from a quantum system by feedback controller is upper bounded by a sum of the decrease of entropy achievable in open-loop control and the mutual information between the quantum system and the controller. This upper bound sets a fundamental limit on the performance of any quantum controllers whose designs are based on the possibilities to attain low entropy states. An application of this approach pertaining to quantum error correction is also discussed.

Multiobjective solution of the uncapacitated plant location problem

In this paper we consider the discrete multiobjective uncapacitated plant location problem. We present an exact and an approximate approach to obtain the set of non-dominated solutions. The two approaches resort to dynamic programming to generate in an efficient way the non-dominated solution sets. The solution methods that solve the problems associated with the generated states are based on the decomposition of the problem on two nested subproblems. We define lower and upper bound sets that lead to elimination tests that have shown to have a high performance. Computational experiments on a set of test problems show the good performance of the proposal.

Information Theoretical Limits on Quantum Control

Fundamental limits on the controllability of quantum mechanical systems are discussed in the light of quantum information theory. It is shown that the amount of entropy-reduction that can be extracted from a quantum system by feedback controller is upper bounded by a sum of the decrease of entropy achievable in open-loop control and the mutual information between the quantum system and the controller. This upper bound sets a fundamental limit on the performance of any quantum controllers whose designs are based on the possibilities to attain low entropy states.

Floquet Boundary Value Problems for Differential Inclusions: a Bound Sets Approach

A technique is developed for the solvability of the Floquet boundary value problem associated to a differential inclusion. It is based on the usage of a not necessarily C^1 -class of Liapunov-like bounding functions. Certain viability arguments are applied for this aim. Some illustrating examples are supplied.

New multivalued functional decomposition algorithms based on MDDs

This paper presents two new functional decomposition partitioning algorithms that use multivalued decision diagrams (MDDs). MDDs are an exceptionally good representation for generalized decomposition because they are canonical and they can represent very large functions. Algorithms developed in this paper are for Boolean/multivalued input and output, completely/incompletely specified functions with application to logic synthesis, machine learning, data mining and knowledge discovery in databases. We compare the run-times and decision diagram sizes of our algorithms to existing decomposition partitioning algorithms based on decision diagrams. The comparisons show that our algorithms are faster and do not result in exponential diagram sizes when decomposing functions with small bound sets.