The homing problem which consists in minimizing the expected time that a controlled one-dimensional diffusion process spends in an interval [ a 1 , a 2 ] is generalized by considering the mathematical expectation of the area between the trajectory of the controlled process and the t-axis from the initial time until the moment it leaves [ a 1 , b 1 ] , where a 1 ≥ 0 . The problem of minimizing the expected value of the square of the first-passage area is also considered. A particular problem is solved explicitly.

In this paper, we investigate a two-person zero-sum semi-Markov games on finite horizon. The performance criterion is the expected-positive-deviation (EPD) of finite horizon payoff from a given level. Under suitable conditions, we not only establish the existence of the value and ϵ ( ≥ 0 ) -saddle points, but also provide an algorithm for computing (at least approximating) the value and ϵ-saddle points. Finally, we analyse the relationship between the EPD criterion and the average-value-at-risk (AVaR) criterion in semi-Markov games.

This paper explores the exponential stability of a fractional stochastic system characterized by hemivariational inequalities and governed by a sectorial operator of order 1<r<2. We begin by formulating the concept of a mild solution using tools from fractional calculus, the theory of sectorial and cosine operators, fixed-point techniques, stochastic analysis, and the generalized Clarke subdifferential. Sufficient conditions are then derived to ensure the exponential decay of the mild solution to zero in the mean-square sense. Furthermore, we demonstrate that the proposed control system attains optimal performance under certain mild conditions. Finally, a representative example is provided to illustrate and validate the theoretical results.

This paper studies the convergence analysis of a relaxed forward-backward-forward splitting algorithm with golden ratio constant momentum for the monotone inclusion problem in Hilbert spaces. The proposed algorithm can be seen as a combination of the forward-backward-forward splitting algorithm with a relaxation term and golden ratio constant momentum. The relaxation term is introduced to enhance convergence flexibility, speed, and robustness, while momentum steps are added for further acceleration. We obtain weak and linear convergence results under standard conditions. Numerical results on signal recovery and image restoration are given to validate the theoretical findings.

In this contribution, variational analysis of constrained optimization over a non-convex set is presented for the obstacle problem. The model describes Timoshenko’s plate with a crack indented by an obstacle given in the form of a curvilinear wedge and subjected to non-penetration constraints. The resulting variational inequality is studied in the weak and strong formulations on the union of convex sets with respect to the existence of a solution, which may be non-unique. The full system of boundary conditions hold at the crack surfaces is derived in the case study of admissible subsets.

This article presents a dynamic algorithm for obtaining the local minimum of an algebraically constrained convex functional. The algorithm is based on the Lagrange multipliers method, and its main advantage is that it converges to the solution in finite time. A differential dynamical system constitutes the mathematical basis for the algorithm's formulation. An experimental methodology is used to validate the proposal. This consists of executing and analysing the algorithm's performance in computing the optimal point of an algebraically constrained convex quadratic functional. Furthermore, the results are numerically compared with the performance of the static form of the Lagrange multipliers method to solve the same problem. Finally, conclusions and future work are presented.

This paper addresses a bilevel optimization problem with a guaranteed (pessimistic) solution in a rather general form. The problem is reduced to a family of bilevel problems with optimistic solutions and perturbed follower's problems. For the quadratic-linear bilevel optimization problem, a reduction of the bilevel problem to a series of non-convex one-level optimization problems is proposed using the penalty method. Additionally, a solution method based on Global Search Theory (GST) is developed for the obtained problem. The GST consists of two stages: special local search and Global Search Scheme (GSS), which is based on Global Optimality Conditions (GOCs). The computational testing of the developed approach on specially constructed bilevel problems of high dimensions was successfully carried out. Finally, the computational testing on specially constructed bilevel problems of rather high dimensions was successfully carried out, that substantiate the developed approach.

In a normed space setting, this paper studies the conditions under which the projected solutions to a quasi equilibrium problem with non-self constraint map exist. Our approach is based on an iterative algorithm which gives rise to a sequence such that, under the assumption of asymptotic regularity, its limit points are projected solutions. Finally, as a particular case, we discuss the existence of projected solutions to a quasi variational inequality problem.

The property HLUR – generalized locally uniformly rotund – has been developed to quantify numerous geometrical behaviours in Banach spaces and their duals. This manuscript investigates the connections between properties (HS) and HLUR with other convexity properties. Additionally, examples in Banach spaces demonstrate the incomparability between HLUR and CLUR Banach spaces. We also discover the relations between these convexity properties that impose the optimality constraints, such as generalized well-posedness. We connect the notion of generalized well-posedness to rotundity properties. We also study the continuity of the pre-duality mappings concerning Banach spaces with the property (HS).

In this paper, we apply our minimax theory ([Ricceri B. On a minimax theorem: an improvement, a new proof and an overview of its applications. Minimax Theory Appl. 2017;2:99–152; Ricceri B. A more complete version of a minimax theorem. Appl Anal Optim. 2021;5:251–261; Ricceri B. Addendum to “A more complete version of a minimax theorem”. Appl Anal Optim. 2022;6:195–197]) with the one developed by A. Moameni in [Moameni A. Critical point theory on convex subsets with applications in differential equations and analysis. J Math Pures Appl. 2020;141:266–315.] to formalize a general scheme giving the multiplicity of critical points. Here is a sample of application of the scheme to a critical elliptic problem: THEOREM. – Let Ω ⊂ R n ( n ≥ 3 ) be a smooth bounded domain and let 1 < q < 2 ≤ p < 2 n n − 2 . Then, for every r , ν > 0 , there exists λ ∗ > 0 with the following property: for every λ ∈ ] 0 , λ ∗ [ , μ ∈ ] − λ ∗ , λ ∗ [ , and for every convex dense set S ⊂ H − 1 ( Ω ) , there exists φ ~ ∈ S , with ‖ φ ~ ‖ H − 1 ( Ω ) < r , such that the problem { − Δu = λ ( | u | 4 n − 2 u + ν | u | q − 2 u + μ | u | p − 2 u + φ ~ ) in Ω u = 0 on ∂Ω has at least two solutions whose norms in H 0 1 ( Ω ) are less than or equal to r.