Optimality of Approximate Quasi-Weakly Efficient Solutions for Vector Equilibrium Problems via Convexificators

This paper is devoted to the investigation of optimality conditions for approximate quasi-weakly efficient solutions to a class of nonsmooth Vector Equilibrium Problem (VEP) via convexificators. First, a necessary optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is presented by making use of the properties of convexificators. Second, the notion of approximate pseudoconvex function in the form of convexificators is introduced, and its existence is verified by a concrete example. Under the introduced generalized convexity assumption, a sufficient optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is also established. Finally, a scalar characterization for approximate quasi-weakly efficient solutions to problem (VEP) is obtained by taking advantage of Tammer’s function.

A Stochastic Maximum Principle for Control Problems Constrained by the Stochastic Navier–Stokes Equations

We analyze the control problem of the stochastic Navier–Stokes equations in multi-dimensional domains considered in Benner and Trautwein (Math Nachr 292(7):1444–1461, 2019) restricted to noise terms defined by a Q-Wiener process. The cost functional related to this control problem is nonconvex. Using a stochastic maximum principle, we derive a necessary optimality condition to obtain explicit formulas the optimal controls have to satisfy. Moreover, we show that the optimal controls satisfy a sufficient optimality condition. As a consequence, we are able to solve uniquely control problems constrained by the stochastic Navier–Stokes equations especially for two-dimensional as well as for three-dimensional domains.

Efficient Algorithms for Rotation Averaging Problems

The rotation averaging problem is a fundamental task in computer vision applications. It is generally very difficult to solve due to the nonconvex rotation constraints. While a sufficient optimality condition is available in the literature, there is a lack of a fast convergent algorithm to achieve stationary points. In this paper, by exploring the problem structure, we first propose a block coordinate descent (BCD)-based rotation averaging algorithm with guaranteed convergence to stationary points. Afterwards, we further propose an alternative rotation averaging algorithm by applying successive upper-bound minimization (SUM) method. The SUM-based rotation averaging algorithm can be implemented in parallel and thus is more suitable for addressing large-scale rotation averaging problems. Numerical examples verify that the proposed rotation averaging algorithms have superior convergence performance as compared to the state-of-the-art algorithm. Moreover, by checking the sufficient optimality condition, we find from extensive numerical experiments that the proposed two algorithms can achieve globally optimal solutions.

Representation of a solution for a system of linear inhomogeneous two-dimensional difference equations of fractional order

One linear inhomogeneous two-parameter discrete fractional system is considered, and the boundary condition is a solution of an analogue of the Cauchy problem for a linear ordinary difference equation. Equation coefficients are given by discrete matrix functions. By introducing an analogue of the Riemann matrix, representations of solutions of the considered boundary value problem are obtained. Note that the result obtained plays an essential role in the linear case for establishing a necessary and sufficient optimality condition in the form of the Pontryagin maximum principle, and also in the general case for studying special control in discrete optimal control problems for systems of 2D fractional orders.

Multiobjective fractional programming problems and the sufficient condition involving Hb – (p, r)-η- invex function

On the basis of arcwise connected convex functions and (p, r) −η - invex functions, we established Hb –(p, r) –η- invex functions. Based on the generalized invex assumption of new functions, the solutions of a class of multiobjective fractional programming problems are studied, and the sufficient optimality condition for the feasible solutions of multiobjective fractional programming problems to be efficient solutions are established and proved.

Geodesic $ \mathcal{E} $-prequasi-invex function and its applications to non-linear programming problems

<p style='text-indent:20px;'>In this article, we define a new class of functions on Riemannian manifolds, called geodesic <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{E} $\end{document}</tex-math></inline-formula>-prequasi-invex functions. By a suitable example it has been shown that it is more generalized class of convex functions. Some of its characteristics are studied on a nonlinear programming problem. We also define a new class of sets, named geodesic slack invex set. Furthermore, a sufficient optimality condition is obtained for a nonlinear programming problem defined on a geodesic local <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{E} $\end{document}</tex-math></inline-formula>-invex set.</p>

Jacobian Nonsingularity in Nonlinear Symmetric Conic Programming Problems and Its Application

This paper considers the nonlinear symmetric conic programming (NSCP) problems. Firstly, a type of strong sufficient optimality condition for NSCP problems in terms of a linear-quadratic term is introduced. Then, a sufficient condition of the nonsingularity of Clarke’s generalized Jacobian of the Karush–Kuhn–Tucker (KKT) system is demonstrated. At last, as an application, this property is used to obtain the local convergence properties of a sequential quadratic programming- (SQP-) type method.

A Second-Order Sufficient Optimality Condition for Risk-Neutral Bi-level Stochastic Linear Programs

The expectation functionals, which arise in risk-neutral bi-level stochastic linear models with random lower-level right-hand side, are known to be continuously differentiable, if the underlying probability measure has a Lebesgue density. We show that the gradient may fail to be local Lipschitz continuous under this assumption. Our main result provides sufficient conditions for Lipschitz continuity of the gradient of the expectation functional and paves the way for a second-order optimality condition in terms of generalized Hessians. Moreover, we study geometric properties of regions of strong stability and derive representation results, which may facilitate the computation of gradients.

Distributed-Order Non-Local Optimal Control

Distributed-order fractional non-local operators were introduced and studied by Caputo at the end of the 20th century. They generalize fractional order derivatives/integrals in the sense that such operators are defined by a weighted integral of different orders of differentiation over a certain range. The subject of distributed-order non-local derivatives is currently under strong development due to its applications in modeling some complex real world phenomena. Fractional optimal control theory deals with the optimization of a performance index functional, subject to a fractional control system. One of the most important results in classical and fractional optimal control is the Pontryagin Maximum Principle, which gives a necessary optimality condition that every solution to the optimization problem must verify. In our work, we extend the fractional optimal control theory by considering dynamical system constraints depending on distributed-order fractional derivatives. Precisely, we prove a weak version of Pontryagin’s maximum principle and a sufficient optimality condition under appropriate convexity assumptions.

Optimality Conditions and Scalarization of Approximate Quasi Weak Efficient Solutions for Vector Equilibrium Problem

This paper is devoted to the investigation of optimality conditions for approximate quasi weak efficient solutions for a class of vector equilibrium problem (VEP). First, a necessary optimality condition for approximate quasi weak efficient solutions to VEP is established by utilizing the separation theorem with respect to the quasirelative interior of convex sets and the properties of the Clarke subdifferential. Second, the concept of approximate pseudoconvex function is introduced and its existence is verified by a concrete example. Under the assumption of introduced convexity, a sufficient optimality condition for VEP in sense of approximate quasi weak efficiency is also presented. Finally, by using Tammer’s function and the directed distance function, the scalarization theorems of the approximate quasi weak efficient solutions of the VEP are proposed.