Mathematics

Based on the global stochastic maximum principle for partially coupled forward-backward stochastic control systems, a modified method of successive approximations (MSA for short) is established for stochastic recursive optimal control problems. The second-order adjoint processes are introduced in the augmented Hamiltonian minimization step in order to find the optimal control which can reach the global minimum of the cost functional. Thanks to the theory of bounded mean oscillation martingales (BMO-martingales for short), we give a delicate proof of the error estimate and obtain the convergence result of the modified MSA algorithm.

We propose new mathematical models of inventory management in a reverse logistics system. The proposed models extend the model introduced by Nahmias and Rivera with the assumption that the demand for newly produced and repaired (remanufacturing) items are not the same. We derive two mathematical models and formulate unconstrained and constrained optimization problems to optimize the holding cost. We also introduce the solution procedures of the proposed problems. The exactness of the proposed solutions has been tested by numerical experiments. Nowadays, it is an essential commitment for industries to reduce greenhouse gas (GHG) emissions as well as energy consumption during the production and remanufacturing processes. This paper also extends along this line of research, and therewith develops a three-objective mathematical model and provides an algorithm to obtain the Pareto solution.

This paper presents a theoretical framework for the design and analysis of gradient descent-based algorithms for coverage control tasks involving robot swarms. We adopt a multiscale approach to analysis and design to ensure consistency of the algorithms in the large-scale limit. First, we represent the macroscopic configuration of the swarm as a probability measure and formulate the macroscopic coverage task as the minimization of a convex objective function over probability measures. We then construct a macroscopic dynamics for swarm coverage, which takes the form of a proximal descent scheme in the L 2 -Wasserstein space. Our analysis exploits the generalized geodesic convexity of the coverage objective function, proving convergence in the L 2 -Wasserstein sense to the target probability measure. We then obtain a consistent gradient descent algorithm in the Euclidean space that is implementable by a finite collection of agents, via a "variational" discretization of the macroscopic coverage objective function. We establish the convergence properties of the gradient descent and its behavior in the continuous-time and large-scale limits. Furthermore, we establish a connection with well-known Lloyd-based algorithms, seen as a particular class of algorithms within our framework, and demonstrate our results via numerical experiments.

This paper proposes a new algorithm -- the \underline{S}ingle-timescale Do\underline{u}ble-momentum \underline{St}ochastic \underline{A}pprox\underline{i}matio\underline{n} (SUSTAIN) -- for tackling stochastic unconstrained bilevel optimization problems. We focus on bilevel problems where the lower level subproblem is strongly-convex and the upper level objective function is smooth. Unlike prior works which rely on \emph{two-timescale} or \emph{double loop} techniques, we design a stochastic momentum-assisted gradient estimator for both the upper and lower level updates. The latter allows us to control the error in the stochastic gradient updates due to inaccurate solution to both subproblems. If the upper objective function is smooth but possibly non-convex, we show that {\aname}~requires O( ϵ ??/2 ) iterations (each using O(1) samples) to find an ϵ -stationary solution. The ϵ -stationary solution is defined as the point whose squared norm of the gradient of the outer function is less than or equal to ϵ . The total number of stochastic gradient samples required for the upper and lower level objective functions matches the best-known complexity for single-level stochastic gradient algorithms. We also analyze the case when the upper level objective function is strongly-convex.

This paper introduces a node formulation for multistage stochastic programs with endogenous (i.e., decision-dependent) uncertainty. Problems with such structure arise when the choices of the decision maker determine a change in the likelihood of future random events. The node formulation avoids an explicit statement of non-anticipativity constraints, and as such keeps the dimension of the model sizeable. An exact solution algorithm for a special case is introduced and tested on a case study. Results show that the algorithm outperforms a commercial solver as the size of the instances increases.

This paper proposes a new parametric level set method for topology optimization based on Deep Neural Network (DNN). In this method, the fully connected deep neural network is incorporated into the conventional level set methods to construct an effective approach for structural topology optimization. The implicit function of level set is described by fully connected deep neural networks. A DNN-based level set optimization method is proposed, where the Hamilton-Jacobi partial differential equations (PDEs) are transformed into parametrized ordinary differential equations (ODEs). The zero-level set of implicit function is updated through updating the weights and biases of networks. The parametrized reinitialization is applied periodically to prevent the implicit function from being too steep or too flat in the vicinity of its zero-level set. The proposed method is implemented in the framework of minimum compliance, which is a well-known benchmark for topology optimization. In practice, designers desire to have multiple design options, where they can choose a better conceptual design base on their design experience. One of the major advantages of DNN-based level set method is its ability to generate diverse and competitive designs with different network architectures. Several numerical examples are presented to verify the effectiveness of the proposed DNN-based level set method.

In many operations management problems, we need to make decisions sequentially to minimize the cost while satisfying certain constraints. One modeling approach to study such problems is constrained Markov decision process (CMDP). When solving the CMDP to derive good operational policies, there are two key challenges: one is the prohibitively large state space and action space; the other is the hard-to-compute transition kernel. In this work, we develop a sampling-based primal-dual algorithm to solve CMDPs. Our approach alternatively applies regularized policy iteration to improve the policy and subgradient ascent to maintain the constraints. Under mild regularity conditions, we show that the algorithm converges at rate O(log(T)/ T ??????) , where T is the number of iterations. When the CMDP has a weakly coupled structure, our approach can substantially reduce the dimension of the problem through an embedded decomposition. We apply the algorithm to two important applications with weakly coupled structures: multi-product inventory management and multi-class queue scheduling, and show that it generates controls that outperform state-of-art heuristics.

We consider a distributed optimal power flow formulated as an optimization problem that maximizes a nondifferentiable concave function. Solving such a problem by the existing distributed algorithms can lead to data privacy issues because the solution information exchanged within the algorithms can be utilized by an adversary to infer the data. To preserve data privacy, in this paper we propose a differentially private projected subgradient (DP-PS) algorithm that includes a solution encryption step. We show that a sequence generated by DP-PS converges in expectation, in probability, and with probability 1. Moreover, we show that the rate of convergence in expectation is affected by a target privacy level of DP-PS chosen by the user. We conduct numerical experiments that demonstrate the convergence and data privacy preservation of DP-PS.

Today, power waste is one of the most crucial problems which power stations across the world are facing. One of the recent trends of the energy system is the lean management technique. When lean management is indicated by the system, customer value is increased and the waste process in industry or in a power station is reduced. In this paper, first of all, we propose mathematical modeling to reduce the cost of production, and then the simulation technique applies to electricity transmission distribution systems. Furthermore, we consider two criteria for comparison including the different costs of the system and the rate of energy waste during the transmission. The primary approach is to use both of the models in order to draw a comparison between simulation results and mathematical models. Finally, the analysis of the test results done by the CPLEX toolbox of MATLAB Software 2019 leads to a remarkable decrease in the costs of energy demand in the electricity transmitting network distribution system.

To guarantee the well-functioning of electricity distribution networks, it is crucial to constantly ensure the demand-supply balance. To do this, one can control the means of production, but also influence the demand: demand-side management becomes more and more popular as the demand keeps increasing and getting more chaotic. In this work, we propose a bilevel model involving an energy supplier and a smart grid operator (SGO): the supplier induces shifts of the load controlled by the SGO by offering time-dependent prices. We assume that the SGO has contracts with consumers and decides their consumption schedule, guaranteeing that the inconvenience induced by the load shifts will not overcome the related financial benefits. Furthermore, we assume that the SGO manages a source of renewable energy (RE), which leads us to consider a stochastic bilevel model, as the generation of RE is by nature highly unpredictable. To cope with the issue of large problem sizes, we design a rolling horizon algorithm that can be applied in a real context.