Short proofs of the universality of certain diagonal quadratic forms

In a paper of Kim, Chan, and Rhagavan, the universal ternary classical quadratic forms over quadratic fields of positive discriminant were discovered. Here a proof of the universality of some of these quadratic forms is given using a technique of Liouville. Another quadratic form over the field of discriminant 8 is shown universal by a different elementary approach. Received: 30 October 2007     

Convex envelopes of separable functions over regions defined by separable functions of the same type

In this paper we derive the convex envelope of separable functions obtained as a linear combination of strictly convex coercive one-dimensional functions over compact regions defined by linear combinations of the same one-dimensional functions. As a corollary of the main result, we are able to derive the convex envelope of any quadratic function (not necessarily separable) over any ellipsoid, and the convex envelope of some quadratic functions over a convex region defined by two quadratic constraints.     

Springer��s theorem for tame quadratic forms over Henselian fields

A quadratic form over a Henselian-valued field of arbitrary residue characteristic is tame if it becomes hyperbolic over a tame extension. The Witt group of tame quadratic forms is shown to be canonically isomorphic to the Witt group of graded quadratic forms over the graded ring associated to the filtration defined by the valuation, hence also isomorphic to a direct sum of copies of the Witt group of the residue field indexed by the value group modulo 2.     

Duality and solutions for quadratic programming over single non-homogeneous quadratic constraint

This paper extends and completes the discussion by Xing et?al. (Canonical dual solutions to the quadratic programming over a quadratic constraint, submitted) about the quadratic programming over one quadratic constraint (QP1QC). In particular, we relax the assumption to cover more general cases when the two matrices from the objective and the constraint functions can be simultaneously diagonalizable via congruence. Under such an assumption, the nonconvex (QP1QC) problem can be solved through a dual approach with no duality gap. This is unusual for general nonconvex programming but we can explain by showing that (QP1QC) is indeed equivalent to a linearly constrained convex problem, which happens to be dual of the dual of itself. Another type of hidden convexity can be also found in the boundarification technique developed in Xing et?al. (Canonical dual solutions to the quadratic programming over a quadratic constraint, submitted).     

A stable range for quadratic forms over commutative rings

A commutative ring A has quadratic stable range 1 (qsr(A) = 1) if each primitive binary quadratic form over A represents a unit. It is shown that qsr(A) = 1 implies that every primitive quadratic form over A represents a unit, A has stable range 1 and finitely generated constant rank projectives over A are free. A classification of quadratic forms is provided over Bezout domains with characteristic other than 2, quadratic stable range 1, and a strong approximation property for a certain subset of their maximum spectrum. These domains include rings of holomorphic functions on connected noncompact Riemann surfaces. Examples of localizations of rings of algebraic integers are provided to show that the classical concept of stable range does not behave well in either direction under finite integral extensions and that qsr(A) = 1 does not descend from such extensions.     

HYERS—ULAM—RASSIAS STABILITY OF QUADRATIC FUNCTIONAL EQUATIONS IN BANACH MODULES OVER A C^＊—ALGEBRA

The authors prove the Hyers-Ulam-Rassias stability of the quadratic mapping in Banach modules over a unital C*-algebra, and prove the Hyers-Ulam-Rassias stability of the quadratic mapping in Banach modules over a unital Banach algebra.     

DC approach to weakly convex optimization and nonconvex quadratic optimization problems

Motivated by weakly convex optimization and quadratic optimization problems, we first show that there is no duality gap between a difference of convex (DC) program over DC constraints and its associated dual problem. We then provide certificates of global optimality for a class of nonconvex optimization problems. As an application, we derive characterizations of robust solutions for uncertain general nonconvex quadratic optimization problems over nonconvex quadratic constraints.     

无界域上不定二次规划的一个算法

本文给出了无界域上不定二次规划一个算法 ,该算法将不定二次规划转化为一系列凸二次规划 ,并证明了算法的收敛性 .     

Robust solutions of quadratic optimization over single quadratic constraint under interval uncertainty

In this paper we examine non-convex quadratic optimization problems over a quadratic constraint under unknown but bounded interval perturbation of problem data in the constraint and develop criteria for characterizing robust (i.e. uncertainty-immunized) global solutions of classes of non-convex quadratic problems. Firstly, we derive robust solvability results for quadratic inequality systems under parameter uncertainty. Consequently, we obtain characterizations of robust solutions for uncertain homogeneous quadratic problems, including uncertain concave quadratic minimization problems and weighted least squares. Using homogenization, we also derive characterizations of robust solutions for non-homogeneous quadratic problems.     

Minimal weakly isotropic forms

In this article weakly isotropic quadratic forms over a (formally) real field are studied. Conditions on the field are given which imply that every weakly isotropic form over that field has a weakly isotropic subform of small dimension. Fields over which every quadratic form can be decomposed into an orthogonal sum of a strongly anisotropic form and a torsion form are characterized in different ways.     

Partial Lagrangian relaxation for general quadratic programming

We give a complete characterization of constant quadratic functions over an affine variety. This result is used to convexify the objective function of a general quadratic programming problem (Pb) which contains linear equality constraints. Thanks to this convexification, we show that one can express as a semidefinite program the dual of the partial Lagrangian relaxation of (Pb) where the linear constraints are not relaxed. We apply these results by comparing two semidefinite relaxations made from two sets of null quadratic functions over an affine variety.      

整体函数域上不可分二次格的构作

构作了有理函数域F_(19)(x)上秩3到6的不可分格,回答了Gerstein关于整体函数域上是否存在秩5的不可分格的问题.     

A geometric characterisation of Desarguesian spreads

We construct and describe the basic properties of a family of semifields in characteristic 2. The construction relies on the properties of projective polynomials over finite fields. We start by associating non-associative products to each such polynomial. The resulting presemifields form the degenerate case of our family. They are isotopic to the Knuth semifields which are quadratic over left and right nuclei. The non-degenerate members of our family display a very different behavior. Their left and right nuclei agree with the center, the middle nucleus is quadratic over the center. None of those semifields is isotopic or Knuth equivalent to a commutative semifield. As a by-product we obtain the complete taxonomy of the characteristic 2 semifields which are quadratic over the middle nucleus, bi-quadratic over the left and right nuclei and not isotopic to twisted fields. This includes determining when two such semifields are isotopic and the order of the autotopism group.     

Sums of squares in function fields of hyperelliptic curves

We study sums of squares, quadratic forms, and related field invariants in a quadratic extension of the rational function field in one variable over a hereditarily pythagorean base field.     

A geometric approach to the quadratic reciprocity law

A geometric viewpoint of much broader potential yields a unified proof of the quadratic reciprocity law based on counting points on quadratic surfaces over finite prime fields.     

Discriminant algebras of finite rank algebras and quadratic trace modules

Based on the construction of the discriminant algebra of an even-ranked quadratic form and Rost’s method of shifting quadratic algebras, we give an explicit rational construction of the discriminant algebra of finite-rank algebras and, more generally, of quadratic trace modules, over arbitrary commutative rings. The discriminant algebra is a tensor functor with values in quadratic algebras, and a symmetric tensor functor with values in quadratic algebras with parity. The automorphism group of a separable quadratic trace module is a smooth, but in general not reductive, group scheme admitting a Dickson type homomorphism into the constant group scheme Z ₂.     

The generalized trust region subproblem: solution complexity and convex hull results

Mathematical Programming - We consider the generalized trust region subproblem (GTRS) of minimizing a nonconvex quadratic objective over a nonconvex quadratic constraint. A lifting of this...     

Approximating Global Quadratic Optimization with Convex Quadratic Constraints

We consider the problem of approximating the global maximum of a quadratic program (QP) subject to convex non-homogeneous quadratic constraints. We prove an approximation quality bound that is related to a condition number of the convex feasible set; and it is the currently best for approximating certain problems, such as quadratic optimization over the assignment polytope, according to the best of our knowledge.     

Regularized Lagrangian duality for linearly constrained quadratic optimization and trust-region problems

In this paper we first establish a Lagrange multiplier condition characterizing a regularized Lagrangian duality for quadratic minimization problems with finitely many linear equality and quadratic inequality constraints, where the linear constraints are not relaxed in the regularized Lagrangian dual. In particular, in the case of a quadratic optimization problem with a single quadratic inequality constraint such as the linearly constrained trust-region problems, we show that the Slater constraint qualification (SCQ) is necessary and sufficient for the regularized Lagrangian duality in the sense that the regularized duality holds for each quadratic objective function over the constraints if and only if (SCQ) holds. A new theorem of the alternative for systems involving both equality constraints and two quadratic inequality constraints plays a key role. We also provide classes of quadratic programs, including a class of CDT-subproblems with linear equality constraints, where (SCQ) ensures regularized Lagrangian duality.