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In this paper, optimality conditions are studied for a class of constrained interval-valued optimization problems governed by path-independent curvilinear integral (mechanical work) cost functionals. Specifically, a minimal criterion of optimality for a local LU-optimal solution of the considered PDE&PDI-constrained variational control problem to be its global LU-optimal solution is formulated and proved. In addition, the main result is highlighted by an illustrative application describing the controlled behavior of an artificial neural system.

Background RNA structure prediction is an important field in bioinformatics, and numerous methods and tools have been proposed. Pseudoknots are specific motifs of RNA secondary structures that are difficult to predict. Almost all existing methods are based on a single model and return one solution, often missing the real structure. An alternative approach would be to combine different models and return a (small) set of solutions, maximizing its quality and diversity in order to increase the probability that it contains the real structure. Results We propose here an original method for predicting RNA secondary structures with pseudoknots, based on integer programming. We developed a generic bi-objective integer programming algorithm allowing to return optimal and sub-optimal solutions optimizing simultaneously two models. This algorithm was then applied to the combination of two known models of RNA secondary structure prediction, namely MEA and MFE. The resulting tool, called BiokoP, is compared with the other methods in the literature. The results show that the best solution (structure with the highest F 1 -score) is, in most cases, given by BiokoP. Moreover, the results of BiokoP are homogeneous, regardless of the pseudoknot type or the presence or not of pseudoknots. Indeed, the F 1 -scores are always higher than 70% for any number of solutions returned. Conclusion The results obtained by BiokoP show that combining the MEA and the MFE models, as well as returning several optimal and several sub-optimal solutions, allow to improve the prediction of secondary structures. One perspective of our work is to combine better mono-criterion models, in particular to combine a model based on the comparative approach with the MEA and the MFE models. This leads to develop in the future a new multi-objective algorithm to combine more than two models. BiokoP is available on the EvryRNA platform: https://EvryRNA.ibisc.univ-evry.fr .

Even deterministic reservoir operation problems with a single objective function may have multiple near‐optimal solutions (MNOS) whose objective values are equal or sufficiently close to the optimum. MNOS is valuable for practical reservoir operation decisions because having a set of alternatives from which to choose allows reservoir operators to explore multiple options whereas the traditional algorithm that produces a single optimum does not offer them this flexibility. This paper presents three methods: the near‐shortest paths (NSP) method, the genetic algorithm (GA) method, and the Markov chain Monte Carlo (MCMC) method, to explore the MNOS. These methods, all of which require a long computation time, find MNOS using different approaches. To reduce the computation time, a narrower subspace, namely a near‐optimal space (NOSP, described by the maximum and minimum bounds of MNOS) is derived. By confining the MNOS search within the NOSP, the computation time of the three methods is reduced. The proposed methods are validated with a test function before they are examined with case studies of both a single reservoir (the Three Gorges Reservoir in China) and a multireservoir system (the Qing River Cascade Reservoirs in China). It is found that MNOS exists for the deterministic reservoir operation problems. When comparing the three methods, the NSP method is unsuitable for large‐scale problems but provides a benchmark to which solutions of small‐ and medium‐scale problems can be compared. The GA method can produce some MNOS but is not very efficient in terms of the computation time. Finally, the MCMC method performs best in terms of goodness‐of‐fit to the benchmark and computation time, since it yields a wide variety of MNOS based on all retained intermediate results as potential MNOS. Two case studies demonstrate that the MNOS identified in this study are useful for real‐world reservoir operation, such as the identification of important operation time periods and tradeoffs among objectives in multipurpose reservoirs. Key Points MNOS within a given value less than the optimal objective do exist MNOS are useful for solving real‐world reservoir operation problems The NSP, GA and MCMC methods, are able to explore MNOS

The new solution concepts of interval-valued multiobjective optimization problems using ordering cone are proposed in this paper. An equivalence relation is introduced to divide the collection of all bounded closed intervals into the equivalence classes. The family of all equivalence classes is also called a quotient set. In this case, this quotient set can turn into a vector space under some suitable settings for vector addition and scalar multiplication. The notions of ordering cone and partial ordering on a vector space are essentially equivalent. It means that an ordering in the quotient set can be defined to study the Pareto optimal solution in multiobjective optimization problems. In this paper, we consider the multiobjective optimization problem such that its coefficients are taken to be the bounded closed intervals. With the help of the convex cone, we can study the Pareto optimal solutions of the multiobjective optimization problem with interval-valued coefficients.

An extended form of robust continuous-time linear programming problem with time-dependent matrices is formulated in this paper. This complicated problem is studied theoretically in this paper. We also design a computational procedure to solve this problem numerically. The desired data that appeared in the problem are considered to be uncertain quantities, which are treated according to the concept of robust optimization. A discretization problem of the robust counterpart is formulated and solved to obtain the ϵ-optimal solutions.

The uncertainty for the continuous-time linear programming problem with time-dependent matrices is considered in this paper. In this case, the robust counterpart of the continuous-time linear programming problem is introduced. In order to solve the robust counterpart, it will be transformed into the conventional form of the continuous-time linear programming problem with time-dependent matrices. The discretization problem is formulated for the sake of numerically calculating the ϵ-optimal solutions, and a computational procedure is also designed to achieve this purpose.

Some new solution concepts to a general fuzzy multiobjective nonlinear programming problem are introduced in this research, and four scalarization techniques are proposed to obtain them. Then, the relation between the set of defined optimal solutions and the set of optimal solutions of the scalarized problems are studied. Moreover, a general scalarized problem is given and shown that these four techniques can be drawn from this problem. Adequate number of numerical examples have been solved to illustrate the techniques.

Optimizing the sum of linear fractional functions over a set of linear inequalities (S-LFP) has been considered by many researchers due to the fact that there are a number of real-world problems which are modelled mathematically as S-LFP problems. Solving the S-LFP is not easy in practice since the problem may have several local optimal solutions which makes the structure complex. To our knowledge, existing methods dealing with S-LFP are iterative algorithms that are based on branch and bound algorithms. Using these methods requires high computational cost and time. In this paper, we present a non-iterative and straightforward method with less computational expenses to deal with S-LFP. In the method, a new S-LFP is constructed based on the membership functions of the objectives multiplied by suitable weights. This new problem is then changed into a linear programming problem (LPP) using variable transformations. It was proven that the optimal solution of the LPP becomes the global optimal solution for the S-LFP. Numerical examples are given to illustrate the method.

For any single-objective mathematical programming model, rank-based optimal solutions are computationally difficult to find compared to an optimal solution to the same single-objective mathematical programming model. In this paper, several methods have been presented to find these rank-based optimal solutions and based on them a new rank-based solution method (RBSM) is outlined to identify non-dominated points set of a multi-objective integer programming model. Each method is illustrated by a numerical example, and for each approach, we have discussed its limitations, advantages and computational complexity.

A robust continuous-time linear programming problem is formulated and solved numerically in this paper. The data occurring in the continuous-time linear programming problem are assumed to be uncertain. In this paper, the uncertainty is treated by following the concept of robust optimization, which has been extensively studied recently. We introduce the robust counterpart of the continuous-time linear programming problem. In order to solve this robust counterpart, a discretization problem is formulated and solved to obtain the ϵ -optimal solution. The important contribution of this paper is to locate the error bound between the optimal solution and ϵ -optimal solution.