Global convergence of augmented Lagrangian methods to a first-order stationary point is well-known to hold under considerably weak constraint qualifications. In particular, several constant rank-type conditions have been introduced for this purpose which turned out to be relevant also beyond this scope. In this paper we show that in fact under these conditions subsequences of … Read more
We present a family of integer programming formulations for the maximum cut problem. These formulations encode the incidence vectors of the cuts of a connected graph by employing a subset of the odd-cycle inequalities that relate to a spanning tree, and they require only the corresponding edge variables to be integral explicitly. They so describe … Read more
In this paper, we present a new method to solve a certain type of Semidefinite Programming (SDP) problems. These types of SDPs naturally arise in the Quadratic Convex Reformulation (QCR) method and can be used to obtain dual bounds of Quadratic Unconstrained Binary Optimization (QUBO) problems. QUBO problems have recently become the focus of attention … Read more
Tai256c is the largest unsolved quadratic assignment problem (QAP) instance in QAPLIB. It is known that QAP tai256c can be converted into a 256 dimensional binary quadratic optimization problem (BQOP) with a single cardinality constraint which requires the sum of the binary variables to be 92. As the BQOP is much simpler than the original … Read more
This work introduces a novel blackbox optimization algorithm for computationally expensive constrained multi-fidelity problems. When applying a direct search method to such problems, the scarcity of feasible points may lead to numerous costly evaluations spent on infeasible points. Our proposed fidelity and interruption controlled optimization algorithm addresses this issue by leveraging multi-fidelity information, allowing for … Read more
The structured saddle-point problem involving the infimal convolution in real Hilbert spaces finds applicability in many applied mathematics disciplines. For this purpose, we develop a stochastic primal-dual splitting algorithm with loopless variance-reduction for solving this generic problem. We first prove the weak almost sure convergence of the iterates. We then demonstrate that our algorithm achieves … Read more
In this paper, we study the low-rank matrix optimization problem, where the penalty term is the $\ell_p~(0<p<1)$ regularization. Inspired by the good performance of half thresholding function in sparse/low-rank recovery problems, we propose a singular value half thresholding (SVHT) algorithm to solve the $\ell_p$ regularized matrix optimization problem. The main iteration in SVHT algorithm makes … Read more
We introduce an autonomous system with closed-loop damping for first-order convex optimization. While, to this day, optimal rates of convergence are only achieved by non-autonomous methods via open-loop damping (e.g., Nesterov’s algorithm), we show that our system is the first one featuring a closed-loop damping while exhibiting a rate arbitrarily close to the optimal one. … Read more
Mixed-integer convex quadratic programs with indicator variables (MIQP) encompass a wide range of applications, from statistical learning to energy, finance, and logistics. The outer approximation (OA) algorithm has been proven efficient in solving MIQP, and the key to the success of an OA algorithm is the strength of the cutting planes employed. In this paper, … Read more
In [R. J. Baraldi and D. P. Kouri, Mathematical Programming, (2022), pp. 1–40], we introduced an inexact trust-region algorithm for minimizing the sum of a smooth nonconvex function and a nonsmooth convex function in Hilbert space—a class of problems that is ubiquitous in data science, learning, optimal control, and inverse problems. This algorithm has demonstrated … Read more