Solving rank-constrained semidefinite programs in exact arithmetic

We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active, this is a non-convex optimization problem, otherwise it is a semidefinite program. Both find numerous applications especially in systems control theory and combinatorial optimization, but even in more general contexts such as polynomial optimization or real algebra. While numerical algorithms exist for solving this problem, such as interior-point or Newton-like algorithms, in this paper we propose an approach based on symbolic computation. We design an exact algorithm for solving rank-constrained semidefinite programs, whose complexity is essentially quadratic on natural degree bounds associated to the given optimization problem: for subfamilies of the problem where the size of the feasible matrix is fixed, the complexity is polynomial in the number of variables. The algorithm works under assumptions on the input data: we prove that these assumptions are generically satisfied. We also implement it in Maple and discuss practical experiments.Comment: Published at ISSAC 2016. Extended version submitted to the Journal of Symbolic Computatio

Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

Optimization is at the core of control theory and appears in several areas of this field, such as optimal control, distributed control, system identification, robust control, state estimation, model predictive control and dynamic programming. The recent advances in various topics of modern optimization have also been revamping the area of machine learning. Motivated by the crucial role of optimization theory in the design, analysis, control and operation of real-world systems, this tutorial paper offers a detailed overview of some major advances in this area, namely conic optimization and its emerging applications. First, we discuss the importance of conic optimization in different areas. Then, we explain seminal results on the design of hierarchies of convex relaxations for a wide range of nonconvex problems. Finally, we study different numerical algorithms for large-scale conic optimization problems.Comment: 18 page

Bad semidefinite programs: they all look the same

Conic linear programs, among them semidefinite programs, often behave pathologically: the optimal values of the primal and dual programs may differ, and may not be attained. We present a novel analysis of these pathological behaviors. We call a conic linear system $Ax <= b$ {\em badly behaved} if the value of $\sup { | A x <= b }$ is finite but the dual program has no solution with the same value for {\em some} $c.$ We describe simple and intuitive geometric characterizations of badly behaved conic linear systems. Our main motivation is the striking similarity of badly behaved semidefinite systems in the literature; we characterize such systems by certain {\em excluded matrices}, which are easy to spot in all published examples. We show how to transform semidefinite systems into a canonical form, which allows us to easily verify whether they are badly behaved. We prove several other structural results about badly behaved semidefinite systems; for example, we show that they are in $NP \cap co-NP$ in the real number model of computing. As a byproduct, we prove that all linear maps that act on symmetric matrices can be brought into a canonical form; this canonical form allows us to easily check whether the image of the semidefinite cone under the given linear map is closed.Comment: For some reason, the intended changes between versions 4 and 5 did not take effect, so versions 4 and 5 are the same. So version 6 is the final version. The only difference between version 4 and version 6 is that 2 typos were fixed: in the last displayed formula on page 6, "7" was replaced by "1"; and in the 4th displayed formula on page 12 "A_1 - A_2 - A_3" was replaced by "A_3 - A_2 - A_1

Semidefinite programming characterization and spectral adversary method for quantum complexity with noncommuting unitary queries

Generalizing earlier work characterizing the quantum query complexity of computing a function of an unknown classical ``black box'' function drawn from some set of such black box functions, we investigate a more general quantum query model in which the goal is to compute functions of N by N ``black box'' unitary matrices drawn from a set of such matrices, a problem with applications to determining properties of quantum physical systems. We characterize the existence of an algorithm for such a query problem, with given error and number of queries, as equivalent to the feasibility of a certain set of semidefinite programming constraints, or equivalently the infeasibility of a dual of these constraints, which we construct. Relaxing the primal constraints to correspond to mere pairwise near-orthogonality of the final states of a quantum computer, conditional on black-box inputs having distinct function values, rather than bounded-error determinability of the function value via a single measurement on the output states, we obtain a relaxed primal program the feasibility of whose dual still implies the nonexistence of a quantum algorithm. We use this to obtain a generalization, to our not-necessarily-commutative setting, of the ``spectral adversary method'' for quantum query lower bounds.Comment: Dagstuhl Seminar Proceedings 06391, "Algorithms and Complexity for Continuous Problems," ed. S. Dahlke, K. Ritter, I. H. Sloan, J. F. Traub (2006), available electronically at http://drops.dagstuhl.de/portals/index.php?semnr=0639

Fast Computation of Common Left Multiples of Linear Ordinary Differential Operators

We study tight bounds and fast algorithms for LCLMs of several linear differential operators with polynomial coefficients. We analyze the arithmetic complexity of existing algorithms for LCLMs, as well as the size of their outputs. We propose a new algorithm that recasts the LCLM computation in a linear algebra problem on a polynomial matrix. This algorithm yields sharp bounds on the coefficient degrees of the LCLM, improving by one order of magnitude the best bounds obtained using previous algorithms. The complexity of the new algorithm is almost optimal, in the sense that it nearly matches the arithmetic size of the output.Comment: The final version will appear in Proceedings of ISSAC 201

Exact duality in semidefinite programming based on elementary reformulations

In semidefinite programming (SDP), unlike in linear programming, Farkas' lemma may fail to prove infeasibility. Here we obtain an exact, short certificate of infeasibility in SDP by an elementary approach: we reformulate any semidefinite system of the form Ai*X = bi (i=1,...,m) (P) X >= 0 using only elementary row operations, and rotations. When (P) is infeasible, the reformulated system is trivially infeasible. When (P) is feasible, the reformulated system has strong duality with its Lagrange dual for all objective functions. As a corollary, we obtain algorithms to generate the constraints of {\em all} infeasible SDPs and the constraints of {\em all} feasible SDPs with a fixed rank maximal solution.Comment: To appear, SIAM Journal on Optimizatio