Mathematics

Robust optimization (RO) is one of the key paradigms for solving optimization problems affected by uncertainty. Two principal approaches for RO, the robust counterpart method and the adversarial approach, potentially lead to excessively large optimization problems. For that reason, first order approaches, based on online-convex-optimization, have been proposed (Ben-Tal et al. (2015), Kilinc-Karzan and Ho-Nguyen (2018)) as alternatives for the case of large-scale problems. However, these methods are either stochastic in nature or involve a binary search for the optimal value. We propose deterministic first-order algorithms based on a saddle-point Lagrangian reformulation that avoids both of these issues. Our approach recovers the other approaches' O(1/epsilon^2) convergence rate in the general case, and offers an improved O(1/epsilon) rate for problems with constraints which are affine both in the decision and in the uncertainty. Experiment involving robust quadratic optimization demonstrates the numerical benefits of our approach.

This work presents a novel version of recently developed Gauss-Newton method for solving systems of nonlinear equations, based on upper bound of solution residual and quadratic regularization ideas. We obtained for such method global convergence bounds and under natural non-degeneracy assumptions we present local quadratic convergence results. We developed stochastic optimization algorithms for presented Gauss-Newton method and justified sub-linear and linear convergence rates for these algorithms using weak growth condition (WGC) and Polyak-Lojasiewicz (PL) inequality. We show that Gauss-Newton method in stochastic setting can effectively find solution under WGC and PL condition matching convergence rate of the deterministic optimization method. The suggested method unifies most practically used Gauss-Newton method modifications and can easily interpolate between them providing flexible and convenient method easily implementable using standard techniques of convex optimization.

Different retail and e-commerce companies are facing the challenge of assembling large numbers of time-critical picking orders that include both single-line and multi-line orders. To reduce unproductive picker working time as in traditional picker-to-parts warehousing systems, different solutions are proposed in the literature and in practice. For example, in a mixed-shelves storage policy, items of the same stock keeping unit are spread over several shelves in a warehouse; or automated guided vehicles (AGVs) are used to transport the picked items from the storage area to packing stations instead of human pickers. This is the first paper to combine both solutions, creating what we call AGV-assisted mixed-shelves picking systems. We model the new integrated order batching and routing problem in such systems as an extended multi-depot vehicle routing problem with both three-index and two-commodity network flow formulations. Due to the complexity of the integrated problem, we develop a novel variable neighborhood search algorithm to solve the integrated problem more efficiently. We test our methods with different sizes of instances, and conclude that the mixed-shelves storage policy is more suitable than the usual storage policy in AGV-assisted mixed-shelves systems for both single-line and multi-line orders (saving up to 67% on driving distances for AGVs). Our variable neighborhood search algorithm provides close-to-optimal solutions within an acceptable computational time.

Multistage stochastic optimization problems are oftentimes formulated informally in a pathwise way. These are correct in a discrete setting and suitable when addressing computational challenges, for example. But the pathwise problem statement does not allow an analysis with mathematical rigor and is therefore not appropriate. This paper addresses the foundations. We provide a novel formulation of multistage stochastic optimization problems by involving adequate stochastic processes as control. The fundamental contribution is a proof that there exist measurable versions of intermediate value functions. Our proof builds on the Kolmogorov continuity theorem. A verification theorem is given in addition, and it is demonstrated that all traditional problem specifications can be stated in the novel setting with mathematical rigor. Further, we provide dynamic equations for the general problem, which is developed for various problem classes. The problem classes covered here include Markov decision processes, reinforcement learning and stochastic dual dynamic programming.

We mainly consider the frequency limited H 2 optimal model order reduction of large-scale sparse generalized systems. For this purpose we need to solve two Sylvester equations. This paper proposes efficient algorithm to solve them efficiently. The ideas are also generalized to index-1 descriptor systems. Numerical experiments are carried out using Python Programming Language and the results are presented to demonstrate the approximation accuracy and computational efficiency of the proposed techniques.

The crossover in solving linear programs is a procedure to recover an optimal corner/extreme point from an approximately optimal inner point generated by interior-point method or emerging first-order methods. Unfortunately it is often observed that the computation time of this procedure can be much longer than the time of the former stage. Our work shows that this bottleneck can be significantly improved if the procedure can smartly take advantage of the problem characteristics and implement customized strategies. For the problem with the network structure, our approach can even start from an inexact solution of interior-point method as well as other emerging first-order algorithms. It fully exploits the network structure to smartly evaluate columns' potential of forming the optimal basis and efficiently identifies a nearby basic feasible solution. For the problem with a large optimal face, we propose a perturbation crossover approach to find a corner point of the optimal face. The comparison experiments with state-of-art commercial LP solvers on classical linear programming problem benchmarks, network flow problem benchmarks and MINST datasets exhibit its considerable advantages in practice.

The projection onto the epigraph or a level set of a closed proper convex function can be achieved by finding a root of a scalar equation that involves the proximal operator as a function of the proximal parameter. This paper develops the variational analysis of this scalar equation. The approach is based on a study of the variational-analytic properties of general convex optimization problems that are (partial) infimal projections of the the sum of the function in question and the perspective map of a convex kernel. When the kernel is the Euclidean norm squared, the solution map corresponds to the proximal map, and thus the variational properties derived for the general case apply to the proximal case. Properties of the value function and the corresponding solution map -- including local Lipschitz continuity, directional differentiability, and semismoothness -- are derived. An SC 1 optimization framework for computing epigraphical and level-set projections is thus established. Numerical experiments on 1-norm projection illustrate the effectiveness of the approach as compared with specialized algorithms

We extend results on backstepping hybrid feedbacks by exploiting synergistic Lyapunov function and feedback (SLFF) pairs in a generalized form. Compared to existing results, we delineate SLFF pairs that are ready-made and do not require extra dynamic variables for backstepping. From an (weak) SLFF pair for an affine control system, we construct an SLFF pair for an extended system where the control input is produced through an integrator. The resulting hybrid feedback asymptotically stabilizes the extended system when the synergy gap for the original system is strictly positive. To highlight the versatility of SLFF pairs, we provide a result on the existence of a SLFF pair whenever a hybrid feedback stabilizer exists. The results are illustrated on the 3D pendulum.

The personalization of cardiac models is the cornerstone of patient-specific modeling. Ideally, non-invasive or minimally-invasive clinical data, such as the standard ECG or intracardiac contact recordings, could provide an insight on model parameters. Parameter selection of such models is however a challenging and potentially time-consuming task. In this work, we estimate the earliest activation sites governing the cardiac electrical activation. Specifically, we introduce GEASI (Geodesic-based Earliest Activation Sites Identification) as a novel approach to simultaneously identify their locations and times. To this end, we start from the anisotropic eikonal equation modeling cardiac electrical activation and exploit its Hamilton-Jacobi formulation to minimize a given objective functional, which in the case of GEASI is the quadratic mismatch to given activation measurements. This versatile approach can be extended for computing topological gradients to estimate the number of sites, or fitting a given ECG. We conducted various experiments in 2D and 3D for in-silico models and an in-vivo intracardiac recording collected from a patient undergoing cardiac resynchronization therapy. The results demonstrate the clinical applicability of GEASI for potential future personalized models and clinical intervention.

In this paper we extend the previous convergence results on the generalized alternating projection method, from subspaces [arXiv:1703.10547], to include smooth manifolds. We show that locally it will behave in the same way, with the same rate as predicted in [arXiv:1703.10547]. The goal is to get closer to a rate for general convex sets, where convergence, but not rate is known. If a finite identification property can be shown for two convex sets, to locally smooth manifolds, then the rates from this paper also apply to those sets. We present a few examples where this is the case, and also a counter example for when this is not the case.