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Discrete optimization belongs to the set of $\mathcal{NP}$-hard problems, spanning fields such as mixed-integer programming and combinatorial optimization. A current standard approach to solving convex discrete optimization problems is the…
We study optimization algorithms for the finite sum problems frequently arising in machine learning applications. First, we propose novel variants of stochastic gradient descent with a variance reduction property that enables linear…
Geometric programming is an important class of optimization problems that enable practitioners to model a large variety of real-world applications, mostly in the field of engineering design. In many real life optimization problem…
We propose an accelerated meta-algorithm, which allows to obtain accelerated methods for convex unconstrained minimization in different settings. As an application of the general scheme we propose nearly optimal methods for minimizing…
We consider rather a general class of multi-level optimization problems, where a convex objective function is to be minimized subject to constraints of optimality of nested convex optimization problems. As a special case, we consider a…
The majorization-minimization (MM) principle is an extremely general framework for deriving optimization algorithms. It includes the expectation-maximization (EM) algorithm, proximal gradient algorithm, concave-convex procedure, quadratic…
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…
To facilitate widespread adoption of automated engineering design techniques, existing methods must become more efficient and generalizable. In the field of topology optimization, this requires the coupling of modern optimization methods…
Computational studies that use block-structured adaptive mesh refinement (AMR) approaches suffer from unnecessarily high mesh resolution in regions adjacent to important solution features. This deficiency limits the performance of AMR…
In this paper, we consider smooth convex optimization problems with simple constraints and inexactness in the oracle information such as value, partial or directional derivatives of the objective function. We introduce a unifying framework,…
Classically, a mainstream approach for solving a convex-concave min-max problem is to instead solve the variational inequality problem arising from its first-order optimality conditions. Is it possible to solve min-max problems faster by…
Multiscale mixed methods based on non-overlapping domain decompositions can efficiently handle the solution of significant subsurface flow problems in very heterogeneous formations of interest to the industry, especially when implemented on…
We study a class of bilevel convex optimization problems where the goal is to find the minimizer of an objective function in the upper level, among the set of all optimal solutions of an optimization problem in the lower level. A wide range…
Probabilistic inference in pairwise Markov Random Fields (MRFs), i.e. computing the partition function or computing a MAP estimate of the variables, is a foundational problem in probabilistic graphical models. Semidefinite programming…
In this paper, we present a novel method for solving multiobjective linear programming problems (MOLPP) that overcomes the need to calculate the optimal value of each objective function. This method is a follow-up to our previous work on…
Hypergraph matching has recently become a popular approach for solving correspondence problems in computer vision as it allows to integrate higher-order geometric information. Hypergraph matching can be formulated as a third-order…
A multi-convex optimization problem is one in which the variables can be partitioned into sets over which the problem is convex when the other variables are fixed. Multi-convex problems are generally solved approximately using variations on…
Many large-scale optimization problems arising in science and engineering are naturally defined at multiple levels of discretization or model fidelity. Multilevel methods exploit this hierarchy to accelerate convergence by combining coarse-…
In a multiobjective optimization problem a solution is called Pareto-optimal if no criterion can be improved without deteriorating at least one of the other criteria. Computing the set of all Pareto-optimal solutions is a common task in…
Variational regularization techniques are dominant in the field of mathematical imaging. A drawback of these techniques is that they are dependent on a number of parameters which have to be set by the user. A by now common strategy to…