This paper describes a new instance library for Quadratic Programming (QP), i.e., the family of continuous and (mixed)-integer optimization problems where the objective function, the constrains, or both are quadratic. QP is a very “varied” class of problems, comprising sub-classes of problems ranging from trivial to undecidable. Solution methods for QP are very diverse, ranging … Read more
Robust model predictive control approaches and other applications lead to nonlinear optimization problems defined on (scenario) trees. We present structure-preserving Quasi-Newton update formulas as well as structured inertia correction techniques that allow to solve these problems by interior-point methods with specialized KKT solvers for tree-structured optimization problems. The same type of KKT solvers could be … Read more
This paper establishes convergence rate bounds for a variant of the proximal alternating direction method of multipliers (ADMM) for solving nonconvex linearly constrained optimization problems. The variant of the proximal ADMM allows the inclusion of an over-relaxation stepsize parameter belonging to the interval (0,2). To the best of our knowledge, all related papers in the … Read more
Motivated by applications in robotics and computer vision, we study problems related to spatial reasoning of a 3D environment using sublevel sets of polynomials. These include: tightly containing a cloud of points (e.g., representing an obstacle) with convex or nearly-convex basic semialgebraic sets, computation of Euclidean distance between two such sets, separation of two convex … Read more
In this paper we consider covariance structural models with which we associate semidefinite programming problems. We discuss statistical properties of estimates of the respective optimal value and optimal solutions when the `true’ covariance matrix is estimated by its sample counterpart. The analysis is based on perturbation theory of semidefinite programming. As an example we consider … Read more
In second-order algorithms, we investigate the relevance of the constant rank of the full set of active constraints in ensuring global convergence to a second-order stationary point. We show that second-order stationarity is not expected in the non-constant rank case if the growth of the so-called tangent multipliers, associated with a second-order complementarity measure, is … Read more
We study in this paper min max robust combinatorial optimization problems for an uncertainty polytope that is defined by knapsack constraints, either in the space of the optimization variables or in an extended space. We provide exact and approximation algorithms that extend the iterative algorithms proposed by Bertismas and Sim (2003). We also study the … Read more
For a given triangle $\triangle ABC$, Pierre de Fermat posed around 1640 the problem of finding a point $P$ minimizing the sum $s_P$ of the Euclidean distances from $P$ to the vertices $A$, $B$, $C$. Based on geometrical arguments this problem was first solved by Torricelli shortly after, by Simpson in 1750, and by several … Read more
The algorithm proposed in this paper runs in a polynomial oracle time, i.e., in a number of arithmetic operations and calls to the separation oracle bounded by a polynomial in the number of variables and in the maximum binary size of an entry of the coefficient matrix. This algorithm is much simpler than traditional polynomial … Read more
A large number of practical algorithms for Optimal Control Problems (OCP) relies on a so-called condensing procedure to exploit the given structure in the quadratic programming (QP) subproblems. While the established structure-exploiting condensing algorithm is of cubic complexity in the horizon length, in this technical note we propose a novel algorithm that is only of … Read more