Related papers: Global Optimization for Cardinality-constrained Mi…
This paper discusses how to find the global minimum of functions that are summations of small polynomials (``small'' means involving a small number of variables). Some sparse sum of squares (SOS) techniques are proposed. We compare their…
Numerous machine learning and industrial problems can be modeled as the minimization of a sum of $N$ so-called clipped convex functions (SCC), i.e. each term of the sum stems as the pointwise minimum between a constant and a convex…
We propose a Branch-and-Cut (B&C) method for solving general MAP-MRF inference problems. The core of our method is a very efficient bounding procedure, which combines scalable semidefinite programming (SDP) and a cutting-plane method for…
Semidefinite programming (SDP) is a fundamental class of convex optimization problems with diverse applications in mathematics, engineering, machine learning, and related disciplines. This paper investigates the application of the…
We consider the global minimization of smooth functions based solely on function evaluations. Algorithms that achieve the optimal number of function evaluations for a given precision level typically rely on explicitly constructing an…
In this paper, Lipschitz univariate constrained global optimization problems where both the objective function and constraints can be multiextremal are considered. The constrained problem is reduced to a discontinuous unconstrained problem…
The stochastic block model (SBM) is a popular tool for community detection in networks, but fitting it by maximum likelihood (MLE) involves a computationally infeasible optimization problem. We propose a new semidefinite programming (SDP)…
We study constrained clustering, where constraints guide the clustering process. In existing works, two categories of constraints have been widely explored, namely pairwise and cardinality constraints. Pairwise constraints enforce the…
In this paper, we consider the problem of partitioning a small data sample of size $n$ drawn from a mixture of $2$ sub-gaussian distributions. Our work is motivated by the application of clustering individuals according to their population…
There are many applications of graph cuts in computer vision, e.g. segmentation. We present a novel method to reformulate the NP-hard, k-way graph partitioning problem as an approximate minimal s-t graph cut problem, for which a globally…
This paper introduces a numerical method to enclose the global minimum of a nonlinear function subject to simple bounds on the variables. Using interval analysis, coupled with the computational power and architecture of graphics processing…
We propose the first general and scalable framework to design certifiable algorithms for robust geometric perception in the presence of outliers. Our first contribution is to show that estimation using common robust costs, such as truncated…
We study how to solve semidefinite programming relaxations for large scale polynomial optimization. When interior-point methods are used, typically only small or moderately large problems could be solved. This paper studies regularization…
Maximum A posteriori Probability (MAP) inference in graphical models amounts to solving a graph-structured combinatorial optimization problem. Popular inference algorithms such as belief propagation (BP) and generalized belief propagation…
Kernel segmentation aims at partitioning a data sequence into several non-overlapping segments that may have nonlinear and complex structures. In general, it is formulated as a discrete optimization problem with combinatorial constraints. A…
We show how to approximate a data matrix $\mathbf{A}$ with a much smaller sketch $\mathbf{\tilde A}$ that can be used to solve a general class of constrained k-rank approximation problems to within $(1+\epsilon)$ error. Importantly, this…
The framework of Integral Quadratic Constraints of Lessard et al. (2014) reduces the computation of upper bounds on the convergence rate of several optimization algorithms to semi-definite programming (SDP). Followup work by Nishihara et…
The 'exact subgraph' approach was recently introduced as a hierarchical scheme to get increasingly tight semidefinite programming relaxations of several NP-hard graph optimization problems. Solving these relaxations is a computational…
In this paper, we propose two algorithms for nonlinear semi-infinite semi-definite programs with infinitely many convex inequality constraints, called SISDP for short. A straightforward approach to the SISDP is to use classical methods for…
We propose a new splitting and successively solving augmented Lagrangian (SSAL) method for solving an optimization problem with both semicontinuous variables and a cardinality constraint. This optimization problem arises in several contexts…