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The corner polyhedron is described by minimal valid inequalities from maximal lattice-free convex sets. For the Relaxed Corner Polyhedron (RCP) with two free integer variables and any number of non-negative continuous variables, it is known…
We construct simple, explicit matrices with columns having unit $\ell^2$ norm and discrepancy approaching $1 + \sqrt{2} \approx 2.414$. This number gives a lower bound, the strongest known as far as we are aware, on the constant appearing…
In this article we derive a doubling inequality at the boundary for solutions to the Kirchhoff-Love isotropic plate's equation satisfying supported boundary conditions. To this end, we combine the use of a suitable conformal mapping which…
Estimation of solution norms and stability for time-dependent nonlinear systems is ubiquitous in numerous engineering, natural science and control problems. Yet, practically valuable results are rare in this area. This paper develops a…
The regret bound of an optimization algorithms is one of the basic criteria for evaluating the performance of the given algorithm. By inspecting the differences between the regret bounds of traditional algorithms and adaptive one, we…
Many practical optimization problems lack strong convexity. Fortunately, recent studies have revealed that first-order algorithms also enjoy linear convergences under various weaker regularity conditions. While the relationship among…
Using techniques of variational analysis, necessary and sufficient subdifferential conditions for H\"older error bounds are investigated and some new estimates for the corresponding modulus are obtained. As an application, we consider the…
The filtering-clustering models, including trend filtering and convex clustering, have become an important source of ideas and modeling tools in machine learning and related fields. The statistical guarantee of optimal solutions in these…
In \cite{cheung2019optimally}, the authors presented two finite element methods for approximating second order boundary value problems on polytopial meshes with optimal accuracy without having to utilize curvilinear mappings. This was done…
Matrix completion results deal with the question of when a partially specified symmetric matrix can be completed to a member of certain matrix cones. Results from positive semidefinite matrix completion and completely positive matrix…
A closed convex conic subset $\mathcal{S}$ of the positive semidefinite (PSD) cone is rank-one generated (ROG) if all of its extreme rays are generated by rank-one matrices. The ROG property of $\mathcal{S}$ is closely related to the…
Roundoff errors cannot be avoided when implementing numerical programs with finite precision. The ability to reason about rounding is especially important if one wants to explore a range of potential representations, for instance for FPGAs…
The uniform quadratic optimizatin problem (UQ) is a nonconvex quadratic constrained quadratic programming (QCQP) sharing the same Hessian matrix. Based on the second-order cone programming (SOCP) relaxation, we establish a new sufficient…
The model problem of a plane angle for a second-order elliptic system subject to Dirichlet, mixed, and Neumann boundary conditions is analyzed. For each boundary condition, the existence of solutions of the form $r^\lambda v$ is reduced to…
We show that kernel-based quadrature rules for computing integrals can be seen as a special case of random feature expansions for positive definite kernels, for a particular decomposition that always exists for such kernels. We provide a…
The theory of error-correcting codes is concerned with constructing codes that optimize simultaneously transmission rate and relative minimum distance. These conflicting requirements determine an asymptotic bound, which is a continuous…
We present a bound for value-prediction error with respect to model misspecification that is tight, including constant factors. This is a direct improvement of the "simulation lemma," a foundational result in reinforcement learning. We…
This paper focuses on developing a conditional gradient algorithm for multiobjective optimization problems with an unbounded feasible region. We employ the concept of recession cone to establish the well-defined nature of the algorithm. The…
Lipschitz continuity of algorithms, introduced by Kumabe and Yoshida (FOCS'23), measures the stability of an algorithm against small input perturbations. Algorithms with small Lipschitz continuity are desirable, as they ensure reliable…
In this paper we investigate necessary conditions of optimality for infinite-horizon optimal control problems with overtaking optimality as an optimality criterion. For the case of local Lipschitz continuity of the payoff function, we…