Applied Mathematics and Optimization最新文献

We provide an Itô’s formula for (C^1)-functionals of flows of conditional marginal distributions of continuous semimartingales. This is based on the notion of weak Dirichlet process, and extends the (C^1)-Itô’s formula in Gozzi and Russo (Stoch Process Appl 116(11):1563–1583, 2006) to this context. As the first application, we study a class of McKean–Vlasov optimal control problems, and establish a verification theorem which only requires (C^1)-regularity of its value function, which is equivalently the (viscosity) solution of the associated HJB master equation. It goes together with a novel duality result.

We prove the existence of solutions for a perturbed differential inclusion governed by a sweeping process with state dependent convex moving set

The novelty brought by our study is the involvement of the Stieltjes derivative (u'_g) with respect to a right-continuous nondecreasing function (g:[0,T]rightarrow {mathbb {R}}), thus establishing a very wide framework containing ODEs, impulsive differential problems, dynamic inclusions on time scales or generalized differential problems. Here (mu _g) is the Stieltjes measure associated to g and (N_{C(t,u(t))}(u(t))) denotes the normal cone of C(t, u(t)) at the point u(t).

A Markov Decision Process (MDP) is a framework used for decision-making. In an MDP problem, the decision maker’s goal is to maximize the expected discounted value of future rewards while navigating through different states controlled by a Markov chain. In this paper, we focus on the case where the transition probabilities vector is deterministic, while the reward vector is uncertain and follow a partially known distribution. We employ a distributionally robust chance constraints approach to model the MDP. This approach entails the construction of potential distributions of reward vector, characterized by moments or statistical metrics. We explore two situations for these ambiguity sets: one where the reward vector has a real support and another where it is constrained to be nonnegative. In the case of a real support, we demonstrate that solving the distributionally robust chance-constrained Markov decision process is mathematically equivalent to a second-order cone programming problem for moments and (phi )-divergence ambiguity sets. For Wasserstein distance ambiguity sets, it becomes a mixed-integer second-order cone programming problem. In contrast, when dealing with nonnegative reward vector, the equivalent optimization problems are different. Moments-based ambiguity sets lead to a copositive optimization problem, while Wasserstein distance-based ambiguity sets result in a biconvex optimization problem. To illustrate the practical application of these methods, we examine a machine replacement problem and present results conducted on randomly generated instances to showcase the effectiveness of our proposed methods.

This work puts forth a new optimal design formulation for planar elastic membranes. The goal is to minimize the membrane’s compliance through choosing the material distribution described by a positive Radon measure. The deformation of the membrane itself is governed by the convexified Föppl’s model. The uniqueness of this model lies in the convexity of its variational formulation despite the inherent nonlinearity of the strain–displacement relation. It makes it possible to rewrite the optimization problem as a pair of mutually dual convex variational problems. The primal variables are displacement functions, whilst in the dual one seeks stresses being Radon measures. The pair of problems is analysed: existence and regularity results are provided, together with the system of optimality criteria. To demonstrate the computational potential of the pair, a finite element scheme is developed around it. Upon reformulation to a conic-quadratic & semi-definite programming problem, the method is employed to produce numerical simulations for several load case scenarios.

This paper is devoted to a Stackelberg stochastic differential game for a linear mean-field type stochastic differential system with a mean-field type quadratic cost functional over a finite horizon. Coefficients in the state equation and weighting matrices in the cost functional are all deterministic. Closed-loop Stackelberg equilibrium strategies are introduced that are independent of initial states. It begins by solving the follower’s stochastic linear quadratic optimal control problem. By transforming the original problem into a new one with a known optimal control, the closed-loop optimal strategy of the follower is characterized by two coupled Riccati equations and a linear mean-field type backward stochastic differential equation. Then the leader turns to solve a stochastic linear quadratic optimal control problem for a mean-field type forward-backward stochastic differential equation. Necessary conditions for the existence of closed-loop optimal strategies for the leader are given by the existence of two coupled Riccati equations with a linear mean-field type backward stochastic differential equation. The solvability of Riccati equations of the leader’s problem is discussed, particularly in cases where the diffusion term of the state equation does not contain the control process of the follower. Moreover, the leader’s value function is expressed via two backward stochastic differential equations and two Lyapunov equations. Finally, a numerical example is given to show the effectiveness of the proposed results.

This paper deals with a nonlocal model for a hyperbolic phase field system coupling the standard energy balance equation for temperature with a dynamic for the phase variable: the latter includes an inertial term and a nonlocal convolution-type operator where the family of kernels depends on a small parameter. We rigorously study the asymptotic convergence of the system as the approximating parameter tends to zero and we obtain at the limit the local system with the elliptic laplacian operator acting on the phase variable. Our analysis is based on some asymptotic properties on nonlocal-to-local convergence that have been recently and successfully applied to families of Cahn–Hilliard models.

This paper is concerned with an optimization problem which is governed by the Kantorovich problem of optimal transport. More precisely, we consider a bilevel optimization problem with the underlying problem being the Kantorovich problem. This task can be reformulated as a mathematical problem with complementarity constraints in the space of regular Borel measures. Because of the non-smoothness that is induced by the complementarity constraints, problems of this type are often regularized, e.g., by an entropic regularization. However, in this paper we apply a quadratic regularization to the Kantorovich problem. By doing so, we are able to drastically reduce its dimension while preserving the sparsity structure of the optimal transportation plan as much as possible. As the title indicates, this is the second part in a series of three papers. While the existence of optimal solutions to both the bilevel Kantorovich problem and its regularized counterpart were shown in the first part, this paper deals with the (weak-(*)) convergence of solutions to the regularized bilevel problem to solutions of the original bilevel Kantorovich problem for vanishing regularization parameters.

This work focuses on moderate deviations for two-time scale systems with mixed fractional Brownian motion. Our proof uses the weak convergence method which is based on the variational representation formula for mixed fractional Brownian motion. Throughout this paper, the Hurst parameter of fractional Brownian motion is larger than 1/2 and the integral along the fractional Brownian motion is understood as the generalized Riemann-Stieltjes integral. First, we consider single-time scale systems with fractional Brownian motion. The key of our proof is showing the weak convergence of the controlled system. Next, we extend our method to show moderate deviations for two-time scale systems. To this goal, we combine the Khasminskii-type averaging principle and the weak convergence approach.

We prove nonuniqueness of weak solutions to multi-dimensional generalisation of the Aw-Rascle model of vehicular traffic. Our generalisation includes the velocity offset in a form of gradient of density function, which results in a dissipation effect, similar to viscous dissipation in the compressible viscous fluid models. We show that despite this dissipation, the extension of the method of convex integration can be applied to generate infinitely many weak solutions connecting arbitrary initial and final states. We also show that for certain choice of data, ill posedness holds in the class of admissible weak solutions.

We investigate the strong convergence properties of a Nesterov type algorithm with two Tikhonov regularization terms in connection to the minimization problem of a smooth convex function f. We show that the generated sequences converge strongly to the minimal norm element from (text {argmin}f). We also show fast convergence for the potential energies (f(x_n)-text {min}f) and (f(y_n)-text {min}f), where ((x_n),,(y_n)) are the sequences generated by our algorithm. Further we obtain fast convergence to zero of the discrete velocity and some estimates concerning the value of the gradient of the objective function in the generated sequences. Via some numerical experiments we show that we need both Tikhonov regularization terms in our algorithm in order to obtain the strong convergence of the generated sequences to the minimum norm minimizer of our objective function.