Let (G, ·) be a 2-divisible group and let (G∗,+) be a uniquely divisible abelian group. We show that the alternative quadratic functional equation f (x y−1)+ f (x y) =±2( f (x)+ f (y)) is equivalent to the quadratic functional equation f (x y−1)+ f (x y) = 2 f (x)+2 f (y) for the class of functions f : G→ G∗.

We classify mutual position of a quadratic Bézier curve and a regular quadric in three dimensional Euclidean space. For given first and last control point, we find the set of all quadratic Bézier curves having no common point with a regular quadric. This system of such quadratic Bézier curves is represented by the set of their admissible middle control points. The spatial problem is reduced to ...

In this note, we establish superiority of the so-called copositive bound over a bound suggested by Nesterov for the quadratic problem to minimize a quadratic form over the l-ball. We illustrate the improvement by simulation results. The copositive bound has the additional advantage that it can be easily extended to the inhomogeneous case of quadratic objectives including a linear term. We also ...

The well-known result stating that any non-convex quadratic problem over the nonnegative orthant with some additional linear and binary constraints can be rewritten as linear problem over the cone of completely positive matrices (Burer, 2009) is generalized by replacing the nonnegative orthant with an arbitrary closed convex cone. This set-semidefinite representation result implies new semidefi...

Solving a quadratic equation P (x) = ax + bx+ c = 0 with real coefficients is known to middle school students. Solving the equation over the quaternions is not straightforward. Huang and So [2] give a complete set of formulas, breaking it into several cases depending on the coefficients. From a result of the second author in [10], zeros of P (x) can be expressed in terms of the zeros of a real ...

This paper presents a bi level large scale multi objective quadratic programming problem (BLLSMOQPP) with fuzzy parameters in the objective functions problem. In the first phase convert the fuzzy parameters of the problem to equivalent crisp problem to make it easy in solving. In the second phase use Taylor series to convert the quadratic problem to linear problem to be easy for solving with th...

In recent years, binary quadratic programming (BQP) has been 7 successively applied to solve several combinatorial optimization problems. We 8 consider in this paper a study of using the BQP model to solve the minimum 9 sum coloring problem (MSCP). For this purpose, we recast the MSCP with 10 a quadratic model which is then solved via a recently proposed Path Relink11 ing (PR) algorithm designe...

We consider nonconvex quadratic optimization problems with binary constraints. Our main result identifies a class of quadratic problems for which a given feasible point is global optimal. We also establish a necessary global optimality condition. These conditions are expressed in a simple way in terms of the problem’s data. We also study the relations between optimal solutions of the nonconvex ...

Kung-Traub’s conjecture states that an optimal iterative method based on d function evaluations for finding a simple zero of a nonlinear function could achieve a maximum convergence order of 2d−1. During the last years, many attempts have been made to prove this conjecture or develop optimal methods which satisfy the conjecture. We understand from the conjecture that the maximum order reached b...

We show in this paper that the roots x1 and x2 of a scalar quadratic polynomial ax2+bx+c = 0 with real or complex coefficients a, b c can be computed in a elementwise mixed stable manner, measured in a relative sense. We also show that this is a stronger property than norm-wise backward stability, but weaker than element-wise backward stability. We finally show that there does not exist any met...