Variational Problems and Applications

1. Introduction

Over the years, many researchers have been interested in obtaining solution procedures in variational (interval/fuzzy) analysis and robust control. In order to formulate necessary and sufficient optimality/efficiency conditions and duality theorems for different classes of robust and interval-valued/fuzzy variational problems, various approaches have been proposed. In this regard, we provide the Special Issue “Variational Problems and Applications” to cover the new advances in these mathematical topics. In this Special Issue, we focused on formulating and demonstrating some characterization results of well-posedness and robust efficient solutions in new classes of (multiobjective) variational (control) problems governed by multiple and/or path-independent curvilinear integral cost functionals and robust mixed and/or isoperimetric constraints involving first- and second-order partial differential equations. In response to our invitation, we received 30 papers from many countries (Romania, China, India, Saudi Arabia, Australia, Egypt, Yemen, Germany, Pakistan, Thailand, Russia), of which 14 were published.

2. Brief Overview of the Contributions

In a review conducted by Treanţă [1], nonlinear dynamics, generated by some classes of constrained control problems that involve second-order partial derivatives, were comprehensively reviewed. Specifically, necessary optimality conditions were formulated and proved for the considered variational control problems governed by integral functionals. In addition, the well-posedness and the associated variational inequalities have been considered in this review paper.

Olteanu [2] briefly reviews a method of approximating any real-valued nonnegative continuous compactly supported function defined on a closed unbounded subset by dominating special polynomials that are sums of squares. This method also works in several-dimensional cases. To perform this, a Hahn–Banach-type theorem (Kantorovich theorem on an extension of positive linear operators), a Haviland theorem, and the notion of a moment-determinate measure were applied. Second, completions and other results of solving full Markov moment problems in terms of quadratic forms are proposed based on polynomial approximation. The existence and uniqueness of the solution are discussed.

Treanţă and Das [3] introduced a new class of multi-dimensional robust optimization problems (named $\left(P\right)$) with mixed constraints, implying second-order partial differential equations (PDEs) and inequations (PDIs). Moreover, they defined an auxiliary (modified) class of robust control problems (named ${\left(P\right)}_{(\overline{b},\overline{c})}$), which is much easier to study, and provided some characterization results of $\left(P\right)$ and ${\left(P\right)}_{(\overline{b},\overline{c})}$ by using the notions of a normal weak robust optimal solution and robust saddle-point associated with a Lagrange functional corresponding to ${\left(P\right)}_{(\overline{b},\overline{c})}$. For this aim, they considered path-independent curvilinear integral cost functionals and the notion of convexity associated with a curvilinear integral functional generated by a controlled closed (complete integrable) Lagrange 1-form.

In 1961, Kestelman first proved the change in the variable theorem for the Riemann integral in its modern form. In 1970, Preiss and Uher supplemented this result with the inverse statement. Later, in a number of papers (Sarkhel, Výborný, Puoso, Tandra, and Torchinsky), the alternative proofs of these theorems were provided within the same formulations. In [4], Kuleshov showed that one of the restrictions (namely, the boundedness of the function f on its entire domain) can be omitted, while the change of variable formula still holds.

By considering the new forms of the notions of lower semicontinuity, pseudomonotonicity, hemicontinuity and monotonicity of the considered scalar multiple integral functional, Treanţă [5] studied the well-posedness of a new class of variational problems with variational inequality constraints. More specifically, by defining the set of approximating solutions for the class of variational problems under study, he established several results on well-posedness.

Guo et al. [6] studied the derivation of optimality conditions and duality theorems for interval-valued optimization problems based on gH-symmetrical derivatives. Further, the concepts of symmetric pseudo-convexity and symmetric quasi-convexity for interval-valued functions are proposed to extend the above optimization conditions. Examples are also presented to illustrate corresponding results.

The concepts of convex and non-convex functions play a key role in the study of optimization. So, with the help of these ideas, some inequalities can also be established. Moreover, the principles of convexity and symmetry are inextricably linked. In the last two years, convexity and symmetry have emerged as a new field due to considerable association. In the work of Khan et al. [7], the authors studied a new version of interval-valued functions (I-V·Fs), known as left and right $\chi$-pre-invex interval-valued functions (LR-$\chi$-pre-invex I-V·Fs). For this class of non-convex I-V·Fs, they derived numerous new dynamic inequalities interval Riemann–Liouville fractional integral operators. The applications of these repercussions are taken into account in a unique way.

Lai et al. [8] introduced a new class of interval-valued preinvex functions termed as harmonically h-preinvex interval-valued functions. They established new inclusion of Hermite–Hadamard for harmonically h-preinvex interval-valued functions via interval-valued Riemann–Liouville fractional integrals. Further, they proved fractional Hermite–Hadamard–type inclusions for the product of two harmonically h-preinvex interval-valued functions. In this way, these findings include several well-known results and newly obtained results of the existing literature as special cases. Moreover, applications of the main results have been demonstrated with some examples.

The principles of convexity and symmetry are inextricably linked. Because of the considerable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In the study of Khan et al. [9], the main aim is to establish the relationship between integral inequalities and interval-valued functions (IV-Fs) based upon the pseudo-order relation. Firstly, we discussed the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs). Then, we obtained a Hermite–Hadamard (H–H) and Hermite–Hadamard–Fejér (H–H–Fejér) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via a pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases have been discussed.

In Alnowibet et al. [10], a hybrid gradient simulated annealing algorithm is guided to solve the constrained optimization problem. When trying to solve constrained optimization problems using deterministic, stochastic optimization methods or using a hybridization between them, penalty function methods are the most popular approach due to their simplicity and ease of implementation. There are many approaches to handling the existence of the constraints in the constraint problem. The simulated-annealing algorithm (SA) is one of the most successful meta-heuristic strategies. On the other hand, the gradient method is the most inexpensive method among the deterministic methods. In previous literature, the hybrid gradient simulated annealing algorithm (GLMSA) demonstrated efficiency and effectiveness in solving unconstrained optimization problems. In Alnowibet et al. [10], the GLMSA algorithm is generalized to solve the constrained optimization problems. Hence, a new approach penalty function is proposed to handle the existence of the constraints. The proposed approach penalty function is used to guide the hybrid gradient simulated annealing algorithm (GLMSA) to obtain a new algorithm (GHMSA) that finds the constrained optimization problem. The performance of the proposed algorithm is tested on several benchmark optimization test problems and some well-known engineering design problems with varying dimensions. Comprehensive comparisons against other methods in the literature are also presented. The results indicate that the proposed method is promising and competitive. The comparison results between the GHMSA and the other four state-Meta-heuristic algorithms indicate that the proposed GHMSA algorithm is competitive with, and in some cases superior to, other existing algorithms in terms of the quality, efficiency, convergence rate, and robustness of the final result.

Data-mining applications are growing with the availability of large data; sometimes, handling large data is also a typical task. Segregation of the data for the extraction of useful information is inevitable for designing modern technologies. Considering this fact, the work of Alrasheedi et al. [11] proposes a chaos-embedded marine predator algorithm (CMPA) for feature selection. The optimization routine is designed with the aim of maximizing the classification accuracy with the optimal number of features selected. The well-known benchmark datasets have been chosen for validating the performance of the proposed algorithm. A comparative analysis of the performance with some well-known algorithms proves the applicability of the proposed algorithm. Further, the analysis was extended to some of the well-known chaotic algorithms; first, the binary versions of these algorithms are developed, and then a comparative analysis of the performance is conducted on the basis of the mean features selected, the classification accuracy obtained and the fitness function values. Statistical significance tests have also been conducted to establish the significance of the proposed algorithm.

In the work of Peng et al. [12], the reverse space-time nonlocal complex modified Kortewewg–de Vries (mKdV) equation is investigated by using the consistent tanh expansion (CTE) method. According to the CTE method, a nonauto-Bäcklund transformation theorem of nonlocal complex mKdV is obtained. The interactions between one kink soliton and other different nonlinear excitations are constructed via the nonauto–Bäcklund transformation theorem. By selecting cnoidal periodic waves, the interaction between one kink soliton and the cnoidal periodic waves is derived. The specific Jacobi function-type solution and graphs of its analysis are provided in this paper.

Lai et al. [13] obtained characterizations of solution sets of the interval-valued mathematical programming problems with switching constraints. Stationary conditions, which are weaker than the standard Karush–Kuhn–Tucker conditions, need to be discussed in order to find the necessary optimality conditions. The authors introduced corresponding weak, Mordukhovich, and strong stationary conditions for the corresponding interval-valued mathematical programming problems with switching constraints (IVPSC) and interval-valued tightened nonlinear problems (IVTNP), because the W-stationary condition of IVPSC is equivalent to the Karush–Kuhn–Tucker conditions of the IVTNP. Furthermore, they used strong stationary conditions to characterize the solution sets for IVTNP, in which the last ones are particular solutions sets for IVPSC, because the feasible set of tightened nonlinear problems (IVTNP) is a subset of the feasible set of the mathematical programs with switching constraints (IVPSC).

In the work of Cipu and Barbu [14], the authors are concerned with solutions for Sturm–Liouville problems (SLP) using a variational problem (VP) formulation of regular SLP. The minimization problem (MP) is also established, and the connection between the solution of each formulation is then proved. Variational estimations (the variational equation associated with the Euler–Lagrange variational principle and Nehari’s method, shooting method and bisection method) and iterative variational methods (He’s method and HPM) for regular RSL are presented in the final part of the paper, which ends with applications.