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Recently, several works have shown that natural modifications of the classical conditional gradient method (aka Frank-Wolfe algorithm) for constrained convex optimization, provably converge with a linear rate when: i) the feasible set is a…
Koopman operators provide tractable means of learning linear approximations of non-linear dynamics. Many approaches have been proposed to find these operators, typically based upon approximations using an a-priori fixed class of models.…
Classical multidimensional scaling only works well when the noisy distances observed in a high dimensional space can be faithfully represented by Euclidean distances in a low dimensional space. Advanced models such as Maximum Variance…
This thesis explores algorithmic applications and limitations of convex relaxation hierarchies for approximating some discrete and continuous optimization problems. - We show a dichotomy of approximability of constraint satisfaction…
We develop and analyze a set of new sequential simulation-optimization algorithms for large-scale multi-dimensional discrete optimization via simulation problems with a convexity structure. The "large-scale" notion refers to that the…
Random projection (RP) is a classical technique for reducing storage and computational costs. We analyze RP-based approximations of convex programs, in which the original optimization problem is approximated by the solution of a…
Nonnegative matrix factorization (NMF) often relies on the separability condition for tractable algorithm design. Separability-based NMF is mainly handled by two types of approaches, namely, greedy pursuit and convex programming. A notable…
Machine learning algorithms in high-dimensional settings are highly susceptible to the influence of even a small fraction of structured outliers, making robust optimization techniques essential. In particular, within the…
Dynamic Programming (DP) provides standard algorithms to solve Markov Decision Processes. However, these algorithms generally do not optimize a scalar objective function. In this paper, we draw connections between DP and (constrained)…
Majorization-minimization algorithms consist of iteratively minimizing a majorizing surrogate of an objective function. Because of its simplicity and its wide applicability, this principle has been very popular in statistics and in signal…
We consider the problem of designing policies for partially observable Markov decision processes (POMDPs) with dynamic coherent risk objectives. Synthesizing risk-averse optimal policies for POMDPs requires infinite memory and thus…
In this paper, we study a class of fractional semi-infinite polynomial programming problems involving s.o.s-convex polynomial functions. For such a problem, by a conic reformulation proposed in our previous work and the quadratic modules…
This paper studies high-dimensional sparse clustering, a combinatorial NP-hard problem arising from the bilinear coupling between cluster assignment and feature selection. We analyze semidefinite programming (SDP) relaxations of $K$-means…
A matrix optimization problem over an uncertain linear system on finite horizon (abbreviated as MOPUL) is studied, in which the uncertain transition matrix is regarded as a decision variable. This problem is in general NP-hard. By using the…
We consider the NP-hard problem of minimizing a convex quadratic function over the integer lattice ${\bf Z}^n$. We present a simple semidefinite programming (SDP) relaxation for obtaining a nontrivial lower bound on the optimal value of the…
Composite convex optimization problems which include both a nonsmooth term and a low-rank promoting term have important applications in machine learning and signal processing, such as when one wishes to recover an unknown matrix that is…
Sparse inverse covariance selection is a fundamental problem for analyzing dependencies in high dimensional data. However, such a problem is difficult to solve since it is NP-hard. Existing solutions are primarily based on convex…
A number of statistical estimation problems can be addressed by semidefinite programs (SDP). While SDPs are solvable in polynomial time using interior point methods, in practice generic SDP solvers do not scale well to high-dimensional…
We consider semidefinite programs (SDPs) of size n with equality constraints. In order to overcome scalability issues, Burer and Monteiro proposed a factorized approach based on optimizing over a matrix Y of size $n$ by $k$ such that $X =…
Quadratic Unconstrained Binary Optimization (QUBO) problems are prevalent in various applications and are known to be NP-hard. The seminal work of Goemans and Williamson introduced a semidefinite programming (SDP) relaxation for such…