We describe a convergence acceleration technique for generic optimization problems. Our scheme computes estimates of the optimum from a nonlinear average of the iterates produced by any optimization method. The weights in this average are computed via a simple linear system, whose solution can be updated online. This acceleration scheme runs in parallel to the … Read more
The need for scalable numerical solutions has motivated the development of asynchronous parallel algorithms, where a set of nodes run in parallel with little or no synchronization, thus computing with delayed information. This paper studies the convergence of the asynchronous parallel algorithm ARock under potentially unbounded delays. ARock is a general asynchronous algorithm that has … Read more
The concept of the optimal partition was originally introduced for linear optimization and linear complementarity problems and subsequently extended to semidefinite optimization. For linear optimization and sufficient linear complementarity problems, from a central solution sufficiently close to the optimal set, the optimal partition and a maximally complementary optimal solution can be identified in strongly polynomial … Read more
The max-$k$-sum of a set of real scalars is the maximum sum of a subset of size $k$, or alternatively the sum of the $k$ largest elements. We study two extensions: First, we show how to obtain smooth approximations to functions that are pointwise max-$k$-sums of smooth functions. Second, we discuss how the max-$k$-sum can … Read more
The paper proposes a linesearch for the primal-dual method. Each iteration of the linesearch requires to update only the dual (or primal) variable. For many problems, in particular for regularized least squares, the linesearch does not require any additional matrix-vector multiplications. We prove convergence of the proposed method under the standard assumptions. We also show … Read more
We consider operators acting on convex subsets of the unit hypercube. These operators are used in constructing convex relaxations of combinatorial optimization problems presented as a 0,1 integer programming problem or a 0,1 polynomial optimization problem. Our focus is mostly on operators that, when expressed as a lift-and-project operator, involve the use of semidefiniteness constraints … Read more
The augmented Lagrangian method (ALM) is fundamental for solving convex minimization models with linear constraints. When the objective function is separable such that it can be represented as the sum of more than one function without coupled variables, various splitting versions of the ALM have been well studied in the literature such as the alternating … Read more
We propose a very simple preprocessing algorithm for semidefinite programming. Our algorithm inspects the constraints of the problem, deletes redundant rows and columns in the constraints, and reduces the size of the variable matrix. It often detects infeasibility. Our algorithm does not rely on any optimization solver: the only subroutine it needs is Cholesky factorization, … Read more
The alternating direction method of multipliers (ADMM) is being widely used for various convex minimization models with separable structures arising in a variety of areas. In the literature, the proximalversion of ADMM which allows ADMM’s subproblems to be proximally regularized has been well studied. Particularly the linearized version of ADMM can be yielded when the … Read more
This paper studies mixed-integer nonlinear programs featuring disjunctive constraints and trigonometric functions. We first characterize the convex hull of univariate quadratic on/off constraints in the space of original variables using perspective functions. We then introduce new tight quadratic relaxations for trigonometric functions featuring variables with asymmetrical bounds. These results are used to further tighten recent … Read more