Optimization and Control (math.OC)

Abstract: The extended Dynamic Mode Decomposition (eDMD) is a very popular method to obtain data-driven surrogate models for nonlinear (control) systems governed by ordinary and stochastic differential equations. Its theoretical foundation is the Koopman framework, in which one propagates observable functions of the state to obtain a linear representation in an infinite-dimensional space. In this work, we prove practical asymptotic stability of a (controlled) equilibrium for eDMD-based model predictive control, in which the optimization step is conducted using the data-based surrogate model. To this end, we derive error bounds that converge to zero if the state approaches the desired equilibrium. Further, we show that, if the underlying system is cost controllable, then this stabilizablility property is preserved. We conduct numerical simulations, which illustrate the proven practical asymptotic stability.

Abstract: Finding a high-quality feasible solution to a combinatorial optimization (CO) problem in a limited time is challenging due to its discrete nature. Recently, there has been an increasing number of machine learning (ML) methods for addressing CO problems. Neural diving (ND) is one of the learning-based approaches to generating partial discrete variable assignments in Mixed Integer Programs (MIP), a framework for modeling CO problems. However, a major drawback of ND is a large discrepancy between the ML and MIP objectives, i.e., variable value classification accuracy over primal bound. Our study investigates that a specific range of variable assignment rates (coverage) yields high-quality feasible solutions, where we suggest optimizing the coverage bridges the gap between the learning and MIP objectives. Consequently, we introduce a post-hoc method and a learning-based approach for optimizing the coverage. A key idea of our approach is to jointly learn to restrict the coverage search space and to predict the coverage in the learned search space. Experimental results demonstrate that learning a deep neural network to estimate the coverage for finding high-quality feasible solutions achieves state-of-the-art performance in NeurIPS ML4CO datasets. In particular, our method shows outstanding performance in the workload apportionment dataset, achieving the optimality gap of 0.45%, a ten-fold improvement over SCIP within the one-minute time limit.

Abstract: Motivated by linear-quadratic optimal control problems (LQ problems, for short) for mean-field stochastic differential equations (SDEs, for short) with the coefficients containing regime switching governed by a Markov chain, we consider an LQ problem for an SDE with the coefficients being adapted to a filtration independent of the Brownian motion driving the control system. Classical approach of completing the square is applied to the current problem and obvious shortcomings are indicated. Open-loop and closed-loop solvability are introduced and characterized.

Abstract: A protein is a sequence of basic blocks called amino acids, and it plays an important role in animals and human beings. The computational protein design (CPD) problem is to identify a protein that could perform some given functions. The CPD problem can be formulated as a quadratic semi-assigement problem (QSAP) and is extremely challenging due to its combinatorial properties over different amino acid sequences. In this paper, we first show that the QSAP is equivalent to its continuous relaxation problem, the RQSAP, in the sense that the QSAP and RQSAP share the same optimal solution. Then we design an efficient quadratic penalty method to solve large-scale RQSAP. Numerical results on benchmark instances verify the superior performance of our approach over the state-of-the-art branch-and-cut solvers. In particular, our proposed algorithm outperforms the state-of-the-art solvers by three order of magnitude in CPU time in most cases while returns a high-quality solution.

Abstract: This paper presents a prescribed performance-based tracking control strategy for the atmospheric reentry flight of space vehicles subject to rapid maneuvers during flight mission. A time-triggered non-monotonic performance funnel is proposed with the aim of constraints violation avoidance in the case of sudden changes of the reference trajectory. Compared with traditional prescribed performance control methods, the novel funnel boundary is adaptive with respect to the reference path and is capable of achieving stability under disturbances. A recursive control structure is introduced which does not require any knowledge of specific system parameters. By a stability analysis we show that the tracking error evolves within the prescribed error margin under a condition which represents a trade-off between the reference signal and the performance funnel. The effectiveness of the proposed control scheme is verified by simulations.

Abstract: The article is devoted to some adaptive methods for variational inequalities with relatively smooth and relatively strongly monotone operators. Starting from the recently proposed proximal variant of the extragradient method for this class of problems, we investigate in detail the method with adaptively selected parameter values. An estimate of the convergence rate of this method is proved. The result is generalized to a class of variational inequalities with relatively strongly monotone generalized smooth variational inequality operators. Numerical experiments have been performed for the problem of ridge regression and variational inequality associated with box-simplex games.

Abstract: Using strategic timetables as input for railway network design has become increasingly popular among western European railway infrastructure operators. Although both railway timetabling and railway network design on their own are well covered by academic research, there is still a gap in the literature concerning timetable-based network design. Therefore, we propose a mixed-integer linear program to design railway infrastructure so that the demand derived from a strategic timetable can be satisfied with minimal infrastructure costs. The demand is given by a list of trains, each featuring start and destination nodes as well as time bounds and a set of frequency and transfer constraints that capture the strategic timetable's main characteristics. During the optimization, the solver decides which railway lines need to be built or expanded and whether travel or headway times must be shortened to meet the demand. Since strategic timetables are subject to uncertainty, we expand the optimization model to a robust version. Uncertain timetables are modelled as discrete scenarios, while uncertain freight train demand is modelled using optional trains, which can be inserted into the resulting timetable if they do not require additional infrastructure. We present computational results for both the deterministic and the robust case and give an outlook on further research.

Abstract: This paper naturally connects the theory of quantum graphs, the control theory and the theory of functional-differential equations. Specifically, we study the problem of damping a control system described by first-order equations on an arbitrary tree graph with global delay. The latter means that the constant delay imposed starting from the initial moment of time propagates through all internal vertices of the graph. By minimizing the energy functional, we arrive at the corresponding variational problem and then prove its equivalence to a self-adjoint boundary value problem on the tree for second-order equations involving both the global delay and the global advance. It is remarkable that the resulting problem acquires Kirchhoff's conditions at the internal vertices of the graph, which often appear in the theory of quantum graphs as well as various applications. The unique solvability of this boundary value problem is proved.

Abstract: The linear conjugate gradient method is an efficient iterative method for the convex quadratic minimization problems $ \mathop {\min }\limits_{x \in { \mathbb R^n}} f(x) =\dfrac{1}{2}x^TAx+b^Tx $, where $ A \in R^{n \times n} $ is symmetric and positive definite and $ b \in R^n $. It is generally agreed that the gradients $ g_k $ are not conjugate with respective to $ A $ in the linear conjugate gradient method (see page 111 in Numerical optimization (2nd, Springer, 2006) by Nocedal and Wright). In the paper we prove the conjugacy of the gradients $ g_k $ generated by the linear conjugate gradient method, namely, $$g_k^TAg_i=0, \; i=0,1,\cdots, k-2.$$ In addition,a new way is exploited to derive the linear conjugate gradient method based on the conjugacy of the search directions and the orthogonality of the gradients, rather than the conjugacy of the search directions and the exact stepsize.

Abstract: Space exploration plans are becoming increasingly complex as public agencies and private companies target deep-space locations, such as cislunar space and beyond, which require long-duration missions and many supporting systems and payloads. Optimizing multi-mission exploration campaigns is challenging due to the large number of required launches as well as their sequencing and compatibility requirements, making the conventional space logistics formulations not scalable. To tackle this challenge, this paper proposes an alternative approach that leverages a two-level hierarchical optimization algorithm: an evolutionary algorithm is used to explore the campaign scheduling solution space, and each of the solutions is then evaluated using a time-expanded multi-commodity flow mixed-integer linear program. A number of case studies, focusing on the Artemis lunar exploration program, demonstrate how the method can be used to analyze potential campaign architectures. The method enables a potential mission planner to study the sensitivity of a campaign to program-level parameters such as logistics vehicle availability and performance, payload launch windows, and in-situ resource utilization infrastructure efficiency.

Abstract: Interior point methods are widely used for different types of mathematical optimization problems. Many implementations of interior point methods in use today rely on direct linear solvers to solve systems of equations in each iteration. The need to solve ever larger optimization problems more efficiently and the rise of hardware accelerators for general purpose computing has led to a large interest in using iterative linear solvers instead, with the major issue being inevitable ill-conditioning of the linear systems arising as the optimization progresses. We investigate the use of Krylov solvers for interior point methods in solving optimization problems from radiation therapy. We implement a prototype interior point method using a so called doubly augmented formulation of the Karush-Kuhn-Tucker (KKT) linear system of equations, originally proposed by Forsgren and Gill, and evaluate its performance on real optimization problems from radiation therapy. Crucially, our implementation uses a preconditioned conjugate gradient method with Jacobi preconditioning internally. Our measurements of the conditioning of the linear systems indicate that the Jacobi preconditioner improves the conditioning of the systems to a degree that they can be solved iteratively, but there is room for further improvement in that regard. Furthermore, profiling of our prototype code shows that it is suitable for GPU acceleration, which may further improve its performance in practice. Overall, our results indicate that our method can find solutions of acceptable accuracy in reasonable time, even with a simple Jacobi preconditioner.

Abstract: Supply chain disruption can occur for a variety of reasons, including natural disasters or market dynamics. If the disruption is profound and with dire consequences for the economy, the regulators may decide to intervene to minimize the impact for the betterment of the society. This paper investigates the minimum quota regulation on transportation amounts, stipulated by the government in a market where transportation capacity is below total production and profitability levels differ significantly among different products. In North America, an interesting example can happen in rail transportation market, where the rail capacity is used for a variety of products and commodities such as oil and grains. This research assumes that there is a shipping company with limited capacity which will ship a group of products with heterogeneous transportation and production costs and prices. Mathematical problems for the market players as well as the government are presented, solutions are proposed, and implemented in a framed Canadian case study. Subsequently, the conditions that justify government intervention are identified, and an algorithm to obtain the optimum minimum quota is presented.