Optimization and Control (math.OC)

Abstract: This paper tackles the problem of finding optimal variable-height transport packaging. The goal is to reduce the empty space left in a box when shipping goods to customers, thereby saving on filler and reducing waste. We cast this problem as a large-scale mixed integer problem (with over seven billion variables) and demonstrate various acceleration techniques to solve it efficiently in about three hours on a laptop. We present a KD-Tree algorithm to avoid exhaustive grid evaluation of the 3D-bin-packing, provide analytical transformations to accelerate the Benders decomposition, and an efficient implementation of the Benders sub problem for significant memory savings and a three order of magnitude runtime speedup.

Abstract: Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control - potentially with non-smoothness both on the level of the objectives or in the system dynamics. This results in new challenges such as dealing with expensive models (e.g., governed by partial differential equations (PDEs)) and developing dedicated algorithms handling the non-smoothness. Since in contrast to single-objective optimization, the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in the field of multiobjective optimization of non-smooth PDE-constrained problems. In particular we report on the advances achieved within Project 2 "Multiobjective Optimization of Non-Smooth PDE-Constrained Problems - Switches, State Constraints and Model Order Reduction" of the DFG Priority Programm 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization".

Abstract: This paper exploits the weak approximation method to study a zero-sum linear differential game under mixed strategies. The stochastic nature of mixed strategies poses challenges in evaluating the game value and deriving the optimal strategies. To overcome these challenges, we first define the mixed strategy based on time discretization given the control period $\delta$. Then, we design a stochastic differential equation (SDE) to approximate the discretized game dynamic with a small approximation error of scale $\mathcal{O}(\delta^2)$ in the weak sense. Moreover, we prove that the game payoff is also approximated in the same order of accuracy. Next, we solve the optimal mixed strategies and game values for the linear quadratic differential games. The effect of the control period is explicitly analyzed when the payoff is a terminal cost. Our results provide the first implementable form of the optimal mixed strategies for a zero-sum linear differential game. Finally, we provide numerical examples to illustrate and elaborate on our results.

Abstract: We propose a novel stochastic smoothing accelerated gradient (SSAG) method for general constrained nonsmooth convex composite optimization, and analyze the convergence rates. The SSAG method allows various smoothing techniques, and can deal with the nonsmooth term that is not easy to compute its proximal term, or that does not own the linear max structure. To the best of our knowledge, it is the first stochastic approximation type method with solid convergence result to solve the convex composite optimization problem whose nonsmooth term is the maximization of numerous nonlinear convex functions. We prove that the SSAG method achieves the best-known complexity bounds in terms of the stochastic first-order oracle ($\mathcal{SFO}$), using either diminishing smoothing parameters or a fixed smoothing parameter. We give two applications of our results to distributionally robust optimization problems. Numerical results on the two applications demonstrate the effectiveness and efficiency of the proposed SSAG method.

Abstract: Public transit plays an essential role in mitigating traffic congestion, reducing emissions, and enhancing travel accessibility and equity. One of the critical challenges in designing public transit systems is distributing finite service supplies temporally and spatially to accommodate time-varying and space-heterogeneous travel demands. Particularly, for regions with low or scattered ridership, there is a dilemma in designing traditional transit lines and corresponding service frequencies. Dense transit lines and high service frequency increase operation costs, while sparse transit lines and low service frequency result in poor accessibility and long passenger waiting time. In the coming era of Mobility-as-a-Service, the aforementioned challenge is expected to be addressed by on-demand services. In this study, we design an On-Demand Multimodel Transit System (ODMTS) for regions with low or scattered travel demands, in which some low-ridership bus lines are replaced with flexible on-demand ride-sharing shuttles. In the proposed ODMTS, riders within service regions can request shuttles to finish their trips or to connect to fixed-route services such as bus, metro, and light rail. Leveraging the integrated transportation system modeling platform, POLARIS, a simulation-based case study is conducted to assess the effectiveness of this system in Austin, Texas.