Set Of Minimal Solutions Research Articles

Minimal solutions of fuzzy relation equations via maximal independent elements

Fuzzy relation equations (FRE) are a useful formalism with a broad number of applications in different computer science areas. Testing if a solution exists and, if so, computing the unique greatest solution is straightforward. In contrast, the computation of minimal solutions is more complex. In particular, even in FRE with a very simple structure, the number of minimal solutions can increase exponentially. However, minimal solutions are immensely useful since, under mild conditions, they (together with the greatest solution) allow one to describe the entire space of solutions to an FRE. The main result of this work is a new method for enumerating the set of minimal solutions. It works by establishing a relationship between coverings of FRE and maximal independent elements of (hyper-)boxes. We can thus make efficient enumeration methods for maximal independent elements of (hyper-)boxes applicable also to our setting of FRE, where the operator considered in the composition of fuzzy relations only needs to preserve suprema of arbitrary subsets and infima of non-empty subsets. More specifically, we thus show that the enumeration of the minimal solutions of an FRE can be done with incremental quasi-polynomial delay.

The stability of minimal solution sets for set optimization problems via improvement sets

In this paper we investigate the stability of solution sets for set optimization problems via improvement sets. Some sufficient conditions for the upper semicontinuity, lower semicontinuity, and compactness of E-minimal solution mappings are given for parametric set optimization under some suitable conditions. We also give some examples to illustrate our main results.

Minimal Solutions of Fuzzy Relation Inequalities With Addition-Min Composition and Their Applications

This paper deals with the set of minimal solutions for fuzzy relation inequalities with addition-min composition. We first show a necessary and sufficient condition for a solution of the fuzzy relation inequalities being a minimal one, then we propose an algorithm to find all minimal solutions of the fuzzy relation inequalities. Finally, we use all minimal solutions of the fuzzy relation inequalities to solve the related optimization problems, which will be illustrated by a series of numeral examples.

Stability of solutions for fuzzy set optimization problems with applications

In this paper, we introduce a new parametric fuzzy set optimization problem based on the ordering relations of fuzzy sets, where a parametric function and an objective function both take values on fuzzy sets. We show the continuity of solution mappings for such a problem under some mild conditions. Moreover, we give the Painlevé-Kuratowski upper convergence and lower convergence of solution sets for fuzzy set optimization problems. As applications, the stability results are applied to obtain the continuity of a minimal solution mapping and the Painlevé-Kuratowski convergence of minimal solution sets for continuous-time fuzzy set optimization problems.

Method for 2D-3D Registration under Inverse Depth and Structural Semantic Constraints for Digital Twin City

A digital twin city maps a virtual three-dimensional (3D) city model to the geographic information system, constructs a virtual world, and integrates real sensor data to achieve the purpose of virtual–real fusion. Focusing on the accuracy problem of vision sensor registration in the virtual digital twin city scene, this study proposes a 2D-3D registration method under inverse depth and structural semantic constraints. First, perspective and inverse depth images of the virtual scene were obtained by using perspective view and inverse-depth nascent technology, and then the structural semantic features were extracted by the two-line minimal solution set method. A simultaneous matching and pose estimation method under inverse depth and structural semantic constraints was proposed to achieve the 2D-3D registration of real images and virtual scenes. The experimental results show that the proposed method can effectively optimize the initial vision sensor pose and achieve high-precision registration in the digital twin scene, and the Z-coordinate error is reduced by 45%. An application experiment of monocular image multi-object spatial positioning was designed, which proved the practicability of this method, and the influence of model data error on registration accuracy was analyzed.

Connectedness of weak minimal solution set for set optimization problems

This paper aims at investigating connectedness of weak p-minimal solution set for set optimization problems by using the scalarization method. We make a new attempt to derive a scalarization result for weak p-minimal solution set for set optimization problems. By using the scalarization result, we establish connectedness of weak p-minimal solution set for set optimization problems.

Stability of efficient solutions to set optimization problems

This article deals with considering stability properties of Pareto minimal solutions to set optimization problems with the set less order relation in real topological Hausdorff vector spaces. We focus on studying the Painleve–Kuratowski convergence of Pareto minimal elements in the image space. Employing convexity properties, we study the external stability of Pareto minimal solutions via weak ones. Then, we use converse properties to investigate external stability conditions to such problems where Pareto minimal solution sets and weak/ideal ones are distinct. For the internal stability, we propose a concept of compact convergence in the sense of Painleve–Kuratowski and use it together with a domination property to analyze stability conditions for the reference problems.

Approximate Solutions and Levitin–Polyak Well-Posedness for Set Optimization Using Weak Efficiency

The present study is devoted to define a new notion of approximate weak minimal solution based on a set order relation introduced by Karaman et al. (Positivity 22(3):783–802, 2018) for a constrained set optimization problem. Sufficient conditions have been found for the closedness of minimal solution sets. Using the Painleve–Kuratowski convergence, the stability aspects of the approximate weak minimal solution sets are discussed. Further, a notion of Levitin–Polyak well-posedness for the set optimization problem is introduced. Sufficiency criteria and some characterizations of the above defined well-posedness are established. An alternative approach to obtain robust solutions for uncertain vector optimization problems is discussed as an application.

Semicontinuity of the Minimal Solution Set Mappings for Parametric Set-Valued Vector Optimization Problems

With the help of a level mapping, this paper mainly investigates the semicontinuity of minimal solution set mappings for set-valued vector optimization problems. First, we introduce a kind of level mapping which generalizes one given in Han and Gong (Optimization 65:1337–1347, 2016). Then, we give a sufficient condition for the upper semicontinuity and the lower semicontinuity of the level mapping. Finally, in terms of the semicontinuity of the level mapping, we establish the upper semicontinuity and the lower semicontinuity of the minimal solution set mapping to parametric set-valued vector optimization problems under the C-Hausdorff continuity instead of the continuity in the sense of Berge.

Stability in unified semi-infinite vector optimization

This paper is devoted to the study of Painleve−Kuratowski convergence of constraint sets, minimal and efficient solution sets of a unified semi-infinite vector optimization problem. The convergence is established under the functional perturbations of objective function and constraint maps assuming the continuous convergence of the objective functions and uniform gamma convergence of the constraint maps. Further, examples are formulated to analyze the significance of assumptions in the main results.

External and internal stability in set optimization

The main aim of this paper is to establish stability in set optimization in terms of convergence of a sequence of solution sets of perturbed set optimization problems to the solution set of the original set optimization problem both in the image space and the given space. The perturbed problems are obtained by perturbing the feasible set without changing the objective map. Formulations of external stability and internal stability are considered in the image space. External stability, which pertains to complete convergence of a subsequence of weak minimal solution sets, both in the sense of Hausdorff and convergence of sets, is established under certain compactness and continuity assumptions. This leads to the upper convergence of the sequence of solution sets in the given space. Internal stability is established for minimal solution sets under certain continuity, compactness and domination assumptions which leads to the lower convergence of the sequence of solution sets in the given space. External stability for minimal solution sets and internal stability for weak minimal solution sets are deduced under the strict quasiconvexity assumption. In particular, the results are also deduced for a vector optimization problem.

Continuity of solutions mappings of parametric set optimization problems

<p style='text-indent:20px;'>Set optimization is an indispensable part of theory and method of optimization, and has been received wide attentions due to its extensive applications in group decision and group game problems. This paper focus on the continuity of the strict (weak) minimal solution set mapping of parametric set-valued vector optimization problems with the lower set less order relation. We firstly introduce a concept of strict lower level mapping of parametric set-valued vector optimization problems. Moreover, the upper and lower semicontinuity of the strict lower level mapping are obtained under some suitable conditions. Lastly, the sufficient condition for the continuity of the strict minimal solution set mappings of parametric set optimization problems are established by a new proof method, which is different from that in [<xref ref-type="bibr" rid="Xu2014">27</xref>,<xref ref-type="bibr" rid="Xu2016">28</xref>].

A solution to the hand-eye calibration in the manner of the absolute orientation problem

PurposeThis paper aims to introduce a simple hand-eye calibration method that can be easily applied with different objective functions.Design/methodology/approachThe hand-eye calibration is solved by using the closed form absolute orientation equations. Instead of processing all samples together, the proposed method goes through all minimal solution sets. Final result is chosen after evaluating the solution set for arbitrary objectives. In this stage, outliers can be excluded optionally if more accuracy is desired.FindingsThe proposed method is very flexible and gives more accurate and convenient results than the existing solutions. The mathematical error expression defined by the calibration equations may not be valid in practice, where especially systematic distortions are present. It is shown in the simulations that the solution which results the least mathematical error in systems may have incorrect, incompatible results in the presence of practical demands.Research limitations/implicationsThe performance of the calibration performed with the proposed method is compared with the reference methods in the literature. When the back-projection error is benchmarked, which corresponds to the point repeatability, the proposed approach is considered as the most successful method among all others. Due to its robustness, it is decided to make tooling-sensor calibrations by the recommended method, in the robotic non-destructive testing station in Ford-OTOSAN Kocaeli Plant Body Shop Department.Originality/valueArranging the well-known AX = XB calibration equation in quaternion representation as Q_A = Q_x × Q_B × Q_x reveals another common spatial rotation equation. In this way, absolute orientation solution satisfies the hand-eye calibration equations. The proposed solution is not presented in the literature as a standalone hand-eye calibration method, although some researchers drop a hint to the relative formulations.

A New Characterisation of the Minimal Solution Set to Max-min Fuzzy Relation Inequalities

In this paper, max-min fuzzy relation inequalities are applied to describe a BitTorrent-like Peer-to-Peer file sharing system. In order to decrease the network congestion under some fixed priority grade of the terminals, we define concept of lexicographic solution in such system. It is found that each minimal solution of max-min fuzzy relation inequalities happen to be a lexicographic solution. Moreover, the lexicographic solution set is exactly the minimal solution set. Numerical example is given to illustrate our results. At last, for the addition-min fuzzy relation inequalities, we provide a guess, trying to describe the relation between its lexicographic solution set and minimal solution set.

Pointwise and global well-posedness in set optimization: a direct approach

The aim of this paper is to characterize some of the pointwise and global well-posedness notions available in literature for a set optimization problem completely by compactness or upper continuity of an appropriate minimal solution set maps. The characterizations of compactness of set-valued maps, lead directly to many characterizations for well-posedness. Sufficient conditions are also given for global well-posedness.

Addition-min fuzzy relation inequalities with application in BitTorrent-like Peer-to-Peer file sharing system

The data transmission mechanism in BitTorrent-like Peer-to-Peer (P2P) file sharing systems may be reduced to some addition-min fuzzy relation inequalities. The solution set of addition-min fuzzy relation inequalities plays an important role in the corresponding optimization problem. In this paper, we study some properties of the solutions to such system. Convexity of the solution set and number of minimal solutions are discussed, with comparison to those of the classical max-T fuzzy relation equations or inequalities. Besides, vertex solution and variable-ordering minimal solution are also investigated, with application in BitTorrent-like P2P file sharing system. Two numerical examples are given to illustrate the feasibility and efficiency of the algorithm for solving the variable-ordering solution.

The Artificial Neural Networks Based on Scalarization Method for a Class of Bilevel Biobjective Programming Problem.

A two-stage artificial neural network (ANN) based on scalarization method is proposed for bilevel biobjective programming problem (BLBOP). The induced set of the BLBOP is firstly expressed as the set of minimal solutions of a biobjective optimization problem by using scalar approach, and then the whole efficient set of the BLBOP is derived by the proposed two-stage ANN for exploring the induced set. In order to illustrate the proposed method, seven numerical examples are tested and compared with results in the classical literature. Finally, a practical problem is solved by the proposed algorithm.

Set optimization using improvement sets

In this paper we introduce a notion of minimal solutions for set-valued optimization problem in terms of improvement sets, by unifying a solution notion, introduced by Kuroiwa [15] for set-valued problems, and a notion of optimal solutions in terms of improvement sets, introduced by Chicco et al. [4] for vector optimization problems. We provide existence theorems for these solutions, and establish lower convergence of the minimal solution sets in the sense of Painlev?-Kuratowski.

Brake orbit solutions for semilinear elliptic systems with asymmetric double well potential

We consider a class of semilinear elliptic system of the form: $$\begin{aligned} -\Delta u(x,y)+\nabla W(u(x,y))=0,\quad (x,y)\in {\mathbb {R}}^{2}, \end{aligned}$$ (0.1) where \(W:{\mathbb {R}}^{2}\rightarrow {\mathbb {R}}\) is a double well potential with minima \(\mathbf{a}_\pm \in {\mathbb {R}}^2\). We show, via variational methods, that if the set of minimal heteroclinic solutions to the one-dimensional system \(-\ddot{q}(x)+\nabla W(q(x))=0,\ x\in {\mathbb {R}}\), up to translations, is finite and constituted by not degenerate functions, then Eq. (0.1) has infinitely many solutions \(u\in C^{2}({\mathbb {R}}^{2})^{2}\), parametrized by an energy value, which are periodic in the variable y and satisfy \(\lim _{x\rightarrow \pm \infty }u(x,y)=\mathbf{a}_{\pm }\) for any \(y\in {\mathbb {R}}\).

Lexicography minimum solution of fuzzy relation inequalities: applied to optimal control in P2P file sharing system

A peer-to-peer (P2P) file sharing system can be reduced into a system of addition-min fuzzy relation inequalities. Concept of lexicography minimum solution is introduced and applied to such system. It is found that the unique lexicography minimum solution can be selected from the minimal solution set of the corresponding fuzzy relation inequalities. However it is difficult to find the minimal solution set which is probably infinite. In order to avoid such difficulty, we propose a so-called Circulation Algorithm to find the unique lexicography minimum solution. The algorithm is developed step by step and illustrated by a numerical application example.