Parameter estimation in nonlinear dynamical systems from observational data is a fundamental inverse problem with applications in many disciplines such as epidemiology, systems biology, climate science, and related fields. In practice, this is further complicated by the fact that observational data are often noisy, sparse, and available only for a subset of the state variables. … Read more
In this paper, we propose and analyze a globally convergent regularized Newton method with positive definite regularization for solving nonsmooth optimization problems. Our approach leverages the coderivative-generated second-order subdifferential (generalized Hessian) and replaces the identity matrix in traditional algorithms with a general positive-definite symmetric matrix to regularize the generalized Hessian. By appropriately selecting the regularization … Read more
Assortment optimization selects a subset of items to maximize expected revenue under a discrete choice model and is widely used in revenue management and online platforms. Its combinatorial nature creates a practical tension among generality, scalability, and provable guarantees: model-specific algorithms can be strong when their structural assumptions hold, but are hard to adapt across … Read more
We consider a general class of “inexactly smooth” convex functions, providing a universal model capturing as special cases $L$-smooth, $M$-Lipschitz, and H\”older smooth functions, and any combination thereof. Such functions possess a calculus closely following that of smooth functions. Our main results provide inexactly smooth functions with interpolation theorems that are necessary and sufficient … Read more
Covering properties build the foundation of stability and sensitivity analysis of solutions to a generalized equation and more specific optimization-related stationarity and equilibrium problems. It has been well-understood that metric regularity of the mapping defining the generalized equation is a key to furnish Lipschitzian stability of the solution of interest. With this work, we want … Read more
We propose and analyze a stochastic direct-search method for unconstrained zeroth-order minimization of locally Lipschitz, possibly nonsmooth, objectives. The method combines random polling directions with a stochastic extrapolating line search based on a sufficient-decrease test of order \(p\). Under conditional accuracy assumptions on the stochastic estimates, which can be verified for mean-zero finite-higher-moment oracle noise … Read more
It is well-known that first-order methods can offer accelerated convergence rates in the presence of growth structures. Restarting schemes provide a general tool for such speed-ups. These schemes typically either require unrealistic problem knowledge, incur logarithmic overhead factors in oracle complexity, and/or have a nontrivial initial burn-in phase. We present a parameter-free approach for restarting … Read more
Shi et al. \cite{shi2022understanding} propose acceleration methods to solve smooth convex optimization problems. In our work, we focus on the general unconstrained composite non-smooth convex optimization problem. We provide an inertial forward-backward algorithm with subgradient correction, derived through time discretization of the ODE, as studied by Shi et al. We achieve the rate of convergence … Read more
In this paper, we study nonsmooth composite optimization problems under heavy-tailed noise, with the objective being a summation of a nested function and a nonsmooth convex regularizer. We propose stochastic proximal approximation methods incorporating a normalization technique to handle the potential challenges caused by the nonsmooth regularizer and heavy-tailed noise. For the case where the … Read more
This work introduces an inexact cubic regularization method with adaptive reuse of Hessian approximations to solve general non-convex optimization problems. In the proposed approach, the gradient is computed inexactly and updated at every iteration, whereas the Hessian approximation is updated at a specific iteration and then reused for $m$ subsequent iterations (a lazy strategy), where … Read more