Convexity of a small ball under quadratic map

SIAM Journal on Optimization, 2006

We describe a Jordan-algebraic version of results related to convexity of images of quadratic mappings as well as related results on exactness of symmetric relaxations of certain classes of nonconvex optimization problems. The exactness of relaxations is proved based on rank estimates. Our approach provides a unifying viewpoint on a large number of classical results related to cones of Hermitian matrices over real and complex numbers. We describe (apparently new) results related to cones of Hermitian matrices with quaternion entries and to the exceptional 27-dimensional Euclidean Jordan algebra.

Optimization Letters, 2012

In this paper we present solutions of two open problems from the theory of quadratic mappings arising from the optimization theory and formulated by Hiriart-Urruty (SIAM Rev 49(2):255-273, 2007). The first problem is concerned with the non-triviality of the set of common zeroes of finitely many of quadratic forms. The second problem deals with the positivity of the maximum of finitely many of quadratic forms. Keywords Quadratic forms • Quadratic mappings • Positive definite quadratic forms In the recent survey paper by Hiriart-Urruty [1] 14 open problems in nonlinear analysis and optimization theory were stated. The thirteenth and fourteenth problems are related to the quadratic mappings theory. Let us recall them. Problem 13 (Solving a System of Quadratic Equations) Let n and k be positive integers, Q i , i = 1,. .. , k, be symmetric n × n matrices. These matrices define quadratic forms q i (x) = Q i x, x over X = R n and a quadratic mapping Q : X → R k by the formula Q(x) = (q 1 (x),. .. , q k (x)). Set K = {x ∈ X : Q(x) = 0, x = 0}. The elements of the cone K are called zeroes of the quadratic mapping Q.

Linear Algebra and its Applications, 2004

The purpose of this paper is to show that the joint numerical range of a m-tuple of n×n hermitian matrices is convex whenever the largest eigenvalue of an associated family of hermitian matrices parameterized by the (m − 1)-dimensional sphere has constant multiplicity and, as a more technical condition, the union over the sphere of the largest eigenvalue eigenspaces does not fill the full n-dimensional complex * Partially supported by the National Science Foundation grant ECS-98-02594. vector space. It is this global, as opposed to local, behavior of the eigenvalues that makes the problem essentially topological. For m ≤ 3, it is shown that the set of hermitian matrices with simple eigenvalues is open and dense in the space of all hermitian matrices, from which it already follows that the numerical range is generically convex for m ≤ 3. From there on, an additional argument shows that convexity always holds when m ≤ 3 and n ≥ 3. Furthermore, our sufficient condition for convexity is in fact a criterion for stable convexity, in the sense that should the sufficient condition fails while convexity holds, the latter can be destroyed by an arbitrarily small perturbation of the data.

Mathematical Programming, 1995

We prove that a general convex quadratic program (QP) can be reduced to the problem of finding the nearest point on a simplicial cone in O(n 3 + n log L) steps, where n and L are, respectively, the dimension and the encoding length of QP. The proof is quite simple and uses duality and repeated perturbation. The implication, however, is nontrivial since the problem of finding the nearest point on a simplicial cone has been considered a simpler problem to solve in the practical sense due to its special structure. Also we show that, theoretically, this reduction implies that (i) if an algorithm solves QP in a polynomial number of elementary arithmetic operations that is independent of the encoding length of data in the objective function then it can be used to solve QP in strongly polynomial time, and (ii) if L is bounded by a 'first order' exponential function of n then (i) can be stated even in stronger terms: to solve QP in strongly polynomial time, it suffices to find an algorithm running in polynomial time that is independent of the encoding length of the quadratic term matrix or constraint matrix. Finally, based on these results, we propose a conjecture.

Statistica Neerlandica, 2002

Let n P 1 and f : R n ! R a convex function. Given distinct points z 1 ; z 2 ; . . . ; z N in R n we consider the problem of finding a quadratic function g : R n ! R such that k½f ðz 1 Þ À gðz 1 Þ; . . . ; f ðz N Þ À gðz N Þk is minimal for a given norm k Á k. For the Euclidean norm this is the wellknown quadratic least squares problem. (If the norm is not specified we will simply refer to g as the quadratic approximation.) In this paper we prove the result that the quadratic approximation is not necessarily convex for n ! 2, even though it is convex if n ¼ 1. This result has many consequences both for the field of statistics and optimization. We show that the best convex quadratic approximation can be obtained in the multivariate case by using semidefinite programming techniques.

In this paper we establish an approximation of approximately quadratic mappings by quadratic mappings, which solves the pertinent Ulam stability problem.

2004

In this paper we study several issues related to the characterization of specific classes of multivariate quadratic mappings that are nonnegative over a given domain, with nonnegativity defined by a pre-specified conic order. In particular, we consider the set (cone) of nonnegative quadratic mappings defined with respect to the positive semidefinite matrix cone, and study when it can be represented by linear matrix inequalities. We also discuss the applications of the results in robust optimization, especially the robust quadratic matrix inequalities and the robust linear programming models. In the latter application the implementational errors of the solution is taken into account, and the problem is formulated as a semidefinite program.

2000

Let A = (A 1 , . . . , A m ) be an m-tuple of n × n Hermitian matrices. For 1 ≤ k ≤ n, the kth joint numerical range of A is defined by

Journal of Inequalities and Applications

In this article, we introduce some special several variables mappings which are quadratic in each variable and show that such mappings can be defined as a single equation that is the generalized multi-quadratic functional equation. We also apply a fixed point theorem to establish the Hyers–Ulam stability for the generalized multi-quadratic functional equations. Furthermore, we present an example and a few corollaries corresponding to some known stability results.

Physica D: Nonlinear Phenomena, 2009

We establish a lower bound on the measure of the set of stable parameters a for the quadratic map Q a (x) = ax(1 − x). For these parameters, we prove that Qa either has a single stable periodic orbit or a perioddoubling bifurcation. From this result, we also obtain a non-trivial upper bound on the set of stochastic parameters for Qa.