15 problems

Spherical quasi-convexity characterization conjecture. This condition is necessary and sufficient to characterize spherically quasi-convex quadratic functions.

Let $\tau_1,\ldots,\tau_n$ be real values, assumed distinct so that the displayed expression is defined, and consider the probability formula for weighted sums of independent unit-…

For any $(v_0,v)\in\mathbb{R}^{n+1}$, let $\text{E-CG}(v_0,v)$ denote the corresponding Eigen-CG inequality, and let BH inequalities denote the Boros–Hammer inequalities. Nonnegati…

Let $\text{Eigen-CG}$ denote the family of Eigen-CG inequalities obtained from vectors $(v_0,v)\in\mathbb{R}^{n+1}$, and let $\text{BH}$ denote the family of Boros–Hammer inequalit…

Computational-cost conjecture. More computational costs are essentially required under the regime $a>2b$.

A CDT subproblem is a cubic-regularized optimization subproblem, and a nonconvex quadratic optimization problem with a quartic constraint is an optimization problem whose objective…

Let … hold; and (ii) there exists a nonzero vector $\bar v$ satisfying $-$. This conjecture seeks a complete characterization of non-strict local non-global minimizers; the supplie…

Single-inequality bound conjecture. The bound in Burer's approach is equal to the best possible bound that can be obtained by adding a single linear inequality.

Let Problem be the general quadratic nonseparable resource allocation problem with generalized bound constraints, and consider its integer version, in which $x_i\in\mathbb{Z}$ for…

Let $x\in\mathbb{R}^2$, $X\in\mathbb{R}^{2\times 2}$, $y\in\mathbb{R}^2$, and let $Y_{12},\alpha\in\mathbb{R}$. Let $\mathcal{H}'$ denote the relevant relaxation in the paper, and…

Quadratic LPL exponent conjecture. The exponent $alpha$ equals either $2$ or $\frac{4}{3}$.

The experiments compare several formulations for quadratic optimization with indicator variables, including formulations with and without cuts, and consider instances whose size is…

Let $V_{\bf f}$ be the variety in the paper's setup, and let $\pi$ be the dimension-preserving morphism from $V_{\bf f} \times \partial_{\rm alg} {\rm S}^{\it ed}_{\bf f}$ onto…

Let $f_0=f_0(x)$ be convex and let $f_1=f_1(x)$ be nonconvex. Define … For $0\leq t\leq 1$, set $f_t:=(1-t)f_0+tf_1$, and suppose that $f_s$ is convex for some $s$ with…

BNR's conjecture. The estimate