Related papers: Duality of convex relaxations for constrained vari…
We propose a solution approach for the problem (P) of minimizing an unconstrained binary polynomial optimization problem. We call this method PQCR (Polynomial Quadratic Convex Reformulation). The resolution is based on a 3-phase method. The…
We discuss a weak constraint qualification for conic linear programs and its applications for a few classes of cones. This constraint qualification is used to give a solution to a problem proposed by Shapiro and Z\v{a}linescu and show that…
In this paper, we introduce a new class of nonsmooth convex functions called SOS-convex semialgebraic functions extending the recently proposed notion of SOS-convex polynomials. This class of nonsmooth convex functions covers many common…
We consider the exact solution of problem $(QP)$ that consists in minimizing a quadratic function subject to quadratic constraints. Starting from the classical convex relaxation that uses the McCormick's envelopes, we introduce 12…
This paper is devoted to the variational inequality problems. We consider two classes of problems, the first is classical constrained variational inequality and the second is the same problem with functional (inequality type) constraints.…
This paper investigates general and generalized differentiation properties of the optimal value function associated with perturbed optimization problems. Fundamental results on nearly convex sets and functions in infinite-dimensional spaces…
This paper studies, for the first time, a bilevel polynomial program whose constraints involve uncertain linear constraints and another uncertain linear optimization problem. In the case of box data uncertainty, we present a sum of squares…
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…
We optimize the running time of the primal-dual algorithms by optimizing their stopping criteria for solving convex optimization problems under affine equality constraints, which means terminating the algorithm earlier with fewer…
In this paper we explore the role of duality principles within the problem of rotation averaging, a fundamental task in a wide range of computer vision applications. In its conventional form, rotation averaging is stated as a minimization…
We investigate exact semidefinite programming (SDP) relaxations for the problem of minimizing a nonconvex quadratic objective function over a feasible region defined by both finitely and infinitely many nonconvex quadratic inequality…
We investigate the convergence of the primal-dual algorithm for composite optimization problems when the objective functions are weakly convex. We introduce a modified duality gap function, which is a lower bound of the standard duality gap…
The training of two-layer neural networks with nonlinear activation functions is an important non-convex optimization problem with numerous applications and promising performance in layerwise deep learning. In this paper, we develop exact…
We develop a general theory of convex duality for certain singular control problems, taking the abstract results by Kramkov and Schachermayer (1999) for optimal expected utility from nonnegative random variables to the level of optimal…
We consider convex optimization problems formulated using dynamic programming equations. Such problems can be solved using the Dual Dynamic Programming algorithm combined with the Level 1 cut selection strategy or the Territory algorithm to…
We revisit the smooth convex-concave bilinearly-coupled saddle-point problem of the form $\min_x\max_y f(x) + \langle y,\mathbf{B} x\rangle - g(y)$. In the highly specific case where each of the functions $f(x)$ and $g(y)$ is either affine…
We propose two new alternating direction methods to solve "fully" nonsmooth constrained convex problems. Our algorithms have the best known worst-case iteration-complexity guarantee under mild assumptions for both the objective residual and…
The main outcomes of the paper are divided into two parts. First, we present a new dual for quadratic programs, in which, the dual variables are affine functions, and we prove strong duality. Since the new dual is intractable, we consider a…
This paper presents the Lagrangian duality theory for mixed-integer semidefinite programming (MISDP). We derive the Lagrangian dual problem and prove that the resulting Lagrangian dual bound dominates the bound obtained from the continuous…
Constrained non-convex optimization is fundamentally challenging, as global solutions are generally intractable and constraint qualifications may not hold. However, in many applications, including safe policy optimization in control and…