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The complexity of Philip Wolfe's method for the minimum Euclidean-norm point problem over a convex polytope has remained unknown since he proposed the method in 1974. The method is important because it is used as a subroutine for one of the…
We develop a principled approach to obtain exact computer-aided worst-case guarantees on the performance of second-order optimization methods on classes of univariate functions. We first present a generic technique to derive interpolation…
Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its…
This paper is concerned with applications of the theory of approximation and interpolation based on compensated convex transforms developed in [K. Zhang, E. Crooks, A. Orlando, Compensated convexity methods for approximations and…
We provide a new method to approximate a (possibly discontinuous) function using Christoffel-Darboux kernels. Our knowledge about the unknown multivariate function is in terms of finitely many moments of the Young measure supported on the…
This paper deals with the kernel-based approximation of a multivariate periodic function by interpolation at the points of an integration lattice -- a setting that, as pointed out by Zeng, Leung, Hickernell (MCQMC2004, 2006) and Zeng,…
It has been well established that first order optimization methods can converge to the maximal objective value of concave functions and provide constant factor approximation guarantees for (non-convex/non-concave) continuous submodular…
Bilevel programming has recently received a great deal of attention due to its abundant applications in many areas. The optimal value function approach provides a useful reformulation of the bilevel problem, but its utility is often limited…
We consider the problem of recovering low-rank matrices from random rank-one measurements, which spans numerous applications including covariance sketching, phase retrieval, quantum state tomography, and learning shallow polynomial neural…
We introduce a definition of a quasiconvex function on an infinite directed regular tree that depends on what we understood by a segment on the tree. Our definition is based on thinking on segments as sub-trees with the root as the midpoint…
Modeling parts of an optimization problem as an optimal value function that depends on a top-level decision variable is a regular occurrence in optimization and an essential ingredient for methods such as Benders Decomposition. It often…
The Performance Estimation Problem (PEP) approach consists in computing worst-case performance bounds on optimization algorithms by solving an optimization problem: one maximizes an error criterion over all initial conditions allowed and…
Solving real-time quadratic programming (QP) is a ubiquitous task in control engineering, such as in model predictive control and control barrier function-based QP. In such real-time scenarios, certifying that the employed QP algorithm can…
We consider the problem of estimating the optimal transport map between two probability distributions, $P$ and $Q$ in $\mathbb R^d$, on the basis of i.i.d. samples. All existing statistical analyses of this problem require the assumption…
The structure uncertainty optimization problem is usually treated as double-loop optimization process, which is computation-intensive. In this paper, an efficient interval uncertainty optimization approach based on Quasi-sparse response…
Matrix completion is a well-studied problem with many machine learning applications. In practice, the problem is often solved by non-convex optimization algorithms. However, the current theoretical analysis for non-convex algorithms relies…
Quasi-Monte Carlo (QMC) methods are applied to multi-level Finite Element (FE) discretizations of elliptic partial differential equations (PDEs) with a random coefficient, to estimate expected values of linear functionals of the solution.…
We extend recent computer-assisted design and analysis techniques for first-order optimization over structured functions--known as performance estimation--to apply to structured sets. We prove "interpolation theorems" for smooth and…
Qubit-efficient optimization studies how large combinatorial problems can be addressed with quantum circuits whose width is far smaller than the number of logical variables. In quadratic unconstrained binary optimization (QUBO), objective…
An index $e$ in a numbering of partial-recursive functions is called minimal if every lesser index computes a different function from $e$. Since the 1960's it has been known that, in any reasonable programming language, no effective…