Option pricing is irreversible, fuzzy, and flexible. The fuzzy measure which is used for real option pricing is a useful supplement to the traditional real option pricing method. Based on the review of the concepts of the mean and variance of trapezoidal fuzzy number and the combination with the Carlsson-Fuller model, the trapezoidal fuzzy variable can be used to represent the current price of land expropriation and the sale price of land on the option day. Fuzzy Black-Scholes option pricing model can be constructed under fuzzy environment and problems also can be solved and discussed through numerical examples.

Let $\left\{A_n\right\}$ be a sequence of nonempty star-shaped sets. By using generalized domination property, we study the lower convergence of minimal sets $\text{Min}{A}_n$. The distinguishing feature of our results lies in disuse of convexity assumptions (only using star-shapedness).

We discuss and develop the convex approximation for robust joint chance constraints under uncertainty of first- and second-order moments. Robust chance constraints are approximated by Worst-Case CVaR constraints which can be reformulated by a semidefinite programming. Then the chance constrained problem can be presented as semidefinite programming. We also find that the approximation for robust joint chance constraints has an equivalent individual quadratic approximation form.

A novel filled function method is suggested for solving box-constrained systems of nonlinear equations. Firstly, the original problem is converted into an equivalent global optimization problem. Subsequently, a novel filled function with one parameter is proposed for solving the converted global optimization problem. Some properties of the filled function are studied and discussed. Finally, an algorithm based on the proposed novel filled function for solving systems of nonlinear equations is presented. The objective function value can be reduced by quarter in each iteration of our algorithm. The implementation of the algorithm on several test problems is reported with satisfactory numerical results.

We introduce and study a class of new general systems of set-valued variational inclusions involving $(A,\eta )$-maximal relaxed monotone operators in Hilbert spaces. By using the general resolvent operator technique associated with $(A,\eta )$-maximal relaxed monotone operators, we construct some new iterative algorithms for finding approximation solutions to the general system of set-valued variational inclusion problem and prove the convergence of this algorithm. Our results improve and extend some known results.

As a major energy-saving industry, power industry has implemented energy-saving generation dispatching. Apart from security and economy, low carbon will be the most important target in power dispatch mechanisms. In this paper, considering a power system with many thermal power generators which use different petrochemical fuels (such as coal, petroleum, and natural gas) to produce electricity, respectively, we establish a self-scheduling model based on the forecasted locational marginal prices, particularly taking into account $\text{C}{\text{O}}_2$ emission constraint, $\text{C}{\text{O}}_2$ emission cost, and unit heat value of fuels. Then, we propose a distributionally robust self-scheduling optimization model under uncertainty in both the distribution form and moments of the locational marginal prices, where the knowledge of the prices is solely derived from historical data. We prove that the proposed robust self-scheduling model can be solved to any precision in polynomial time. These arguments are confirmed in a practical example on the IEEE 30 bus test system.

We introduce and analyze a hybrid iterative algorithm by virtue of Korpelevich's extragradient method, viscosity approximation method, hybrid steepest-descent method, and averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inequality problems (VIPs), the solution set of general system of variational inequalities (GSVI), and the set of minimizers of convex minimization problem (CMP), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others.

An interior projected-like subgradient method for mixed variational inequalities is proposed in finite dimensional spaces, which is based on using non-Euclidean projection-like operator. Under suitable assumptions, we prove that the sequence generated by the proposed method converges to a solution of the mixed variational inequality. Moreover, we give the convergence estimate of the method. The results presented in this paper generalize some recent results given in the literatures.

This paper presents a branch and bound algorithm for globally solving the sum of concave-convex ratios problem (P) over a compact convex set. Firstly, the problem (P) is converted to an equivalent problem (P1). Then, the initial nonconvex programming problem is reduced to a sequence of convex programming problems by utilizing linearization technique. The proposed algorithm is convergent to a global optimal solution by means of the subsequent solutions of a series of convex programming problems. Some examples are given to illustrate the feasibility of the proposed algorithm.

Firstly, a two-unit cold standby shock model with multiple adaptive vacations is introduced, in which the startup and replacement of repair facility are also considered. Secondly, using supplementary variable method and Laplace transform, some important reliability indices are derived, such as availability, failure frequency, mean vacation period, mean renewal cycle, mean startup period, and replacement frequency. Finally, a production line controlled by two cold-standby computers is modeled to present numerical illustration and its optimal part-time job policy at a maximum profit.

We consider the linear programming problem with uncertainty set described by $\left(p,w\right)$-norm. We suggest that the robust counterpart of this problem is equivalent to a computationally convex optimization problem. We provide probabilistic guarantees on the feasibility of an optimal robust solution when the uncertain coefficients obey independent and identically distributed normal distributions.

We introduce and analyze one iterative algorithm by hybrid shrinking projection method for finding a solution of the minimization problem for a convex and continuously Fréchet differentiable functional, with constraints of several problems: finitely many generalized mixed equilibrium problems, finitely many variational inequalities, the general system of variational inequalities and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another iterative algorithm by hybrid shrinking projection method for finding a fixed point of infinitely many nonexpansive mappings with the same constraints, and derive its strong convergence under mild assumptions.

From the on-line point, we consider the hospital’s medical examination appointment problem with hierarchical machines. This approach eliminates the need for both demand forecasts and a risk-neutrality assumption. Due to different unit revenue, uncertain demand, and arrival of patients, we design on-line booking policies for two kinds of different situations from the perspective of on-line policy and competitive analysis. After that, we prove the optimal competitive ratios. Through numerical examples, we compare advantages and disadvantages between on-line policies and traditional policies, finding that there is different superiority for these two policies under different arrival sequences.

We study Mann type iterative algorithms for finding fixed points of Bregman relatively nonexpansive mappings in Banach spaces. By exhibiting an example, we first show that the class of Bregman relatively nonexpansive mappings embraces properly the class of Bregman strongly nonexpansive mappings which was investigated by Martín-Márques et al. (2013). We then prove weak convergence theorems for the sequences produced by the methods. Some application of our results to the problem of finding a zero of a maximal monotone operator in a Banach space is presented. Our results improve and generalize many known results in the current literature.

A class of vector optimization problems is considered and a characterization of $E$-Benson proper efficiency is obtained by using a nonlinear scalarization function proposed by Göpfert et al. Some examples are given to illustrate the main results.

By using slack variables and minimum function, we first reformulate the system of equalities and inequalities as a system of nonsmooth equations, and, using smoothing technique, we construct the smooth operator. A new noninterior continuation method is proposed to solve the system of smooth equations. It shows that any accumulation point of the iteration sequence generated by our algorithm is a solution of the system of equalities and inequalities. Some numerical experiments show the feasibility and efficiency of the algorithm.

Some necessary global optimality conditions and sufficient global optimality conditions for nonconvex minimization problems with a quadratic inequality constraint and a linear equality constraint are derived. In particular, global optimality conditions for nonconvex minimization over a quadratic inequality constraint which extend some known global optimality conditions in the existing literature are presented. Some numerical examples are also given to illustrate that a global minimizer satisfies the necessary global optimality conditions but a local minimizer which is not global may fail to satisfy them.

We study a general second-order $m$-point boundary value problems for nonlinear singular impulsive dynamic equations on time scales $u^{\Delta \nabla }(t)+a(t)u^{\Delta }(t)+b(t)u(t)+q(t)f(t,u(t))=\mathrm{0},t\in (\mathrm{0,1}),t\ne t_k,u^{\Delta }(t_k^{+})=u^{\Delta }(t_k)-I_k(u(t_k))$, and $k=\mathrm{1,2},\dots ,n,\mathrm{}u(\rho (\mathrm{0}))=\mathrm{0},u(\sigma (\mathrm{1}))={\sum }_{i=\mathrm{1}}^{m-\mathrm{2}}{\alpha }_{i}u({\eta }_i)\mathrm{‍}.$ The existence and uniqueness of positive solutions are established by using the mixed monotone fixed point theorem on cone and Krasnosel’skii fixed point theorem. In this paper, the function items may be singular in its dependent variable. We present examples to illustrate our results.

We mainly discuss the properties and applications of semi-$G$-preinvexity, which was introduced in Peng and Chang 2013. Examples are given to show the existence of semi-$G$-preinvex functions. Three important properties and characterizations of semi-$G$-preinvex functions are established. Furthermore, under semi-$G$-preinvexity, some optimality results are obtained in nonlinear programming problems with inequality constraint. Our results improve and generalize some known results.

A grey game model is constructed for enterprise carbon emissions reduction and government policies on energy conservation, which explains the interaction equilibrium strategies between government and enterprises with resource constraints. The MATLAB tools are used to simulate the game process, and the result shows that the optimal strategy is consistent with the operation result. Finally, macroscopic countermeasures and suggestions are proposed for government and enterprise.

We introduce a class of functions called geodesic $B$-preinvex and geodesic $B$-invex functions on Riemannian manifolds and generalize the notions to the so-called geodesic quasi/pseudo $B$-preinvex and geodesic quasi/pseudo $B$-invex functions. We discuss the links among these functions under appropriate conditions and obtain results concerning extremum points of a nonsmooth geodesic $B$-preinvex function by using the proximal subdifferential. Moreover, we study a differentiable multiobjective optimization problem involving new classes of generalized geodesic $B$-invex functions and derive Kuhn-Tucker-type sufficient conditions for a feasible point to be an efficient or properly efficient solution. Finally, a Mond-Weir type duality is formulated and some duality results are given for the pair of primal and dual programming.

We modify the relaxed hybrid steepest-descent methods to the case of variational inequality for finding a solution over the set of common fixed points of a finite family of strictly pseudocontractive mappings. The strongly monotone property defined on cost operator was extended to relaxed cocoercive in convergence analysis. Results presented in this paper may be viewed as a refinement and important generalizations of the previously known results announced by many other authors.

This paper is mainly devoted to the study of implicit multifunction theorems in terms of Clarke coderivative in general Banach spaces. We present new sufficient conditions for the local metric regularity, metric regularity, Lipschitz-like property, nonemptiness, and lower semicontinuity of implicit multifunctions in general Banach spaces. The basic tools of our analysis involve the Ekeland variational principle, the Clarke subdifferential, and the Clarke coderivative.