56 problems

Upper-bound conjecture. The classical Hamiltonian satisfies the bound

A convex semialgebraic set is a semialgebraic subset of some finite-dimensional real vector space that is convex, and a spectrahedral shadow is the image of a spectrahedron under a…

Let $H_n=[0,1]^n$ be the hypercube, described by $g_i=x_i-x_i^2\geq 0$ for $i=1,\ldots,n$. For even $n$, let $M_{n,n}(\mathbf g)$ be the degree-$n$ truncated quadratic module gener…

Let $P_{\mathbb{P}^n,2d}$ be the cone of globally nonnegative forms of degree $2d$ on $\mathbb{P}^n$. Extreme-ray conjecture. Every globally nonnegative form spanning an extreme ra…

Let a biquadratic form be a homogeneous polynomial of degree two in each of two variable vectors, and let its SOS rank be the smallest number of squares of bilinear forms in a sum-…

For integers $p$ and $q$ with $0<q<p$, let $N(q/p)$ denote the number of elements in a minimal $(0,p)$-mediated sequence containing $q$. A sequence of integers…

Let $X\subset \mathbb{P}^n$ be a totally real variety. No-gaps conjecture for stubborn degrees. There exists a positive integer $d$ such that for every integer $k$ with $1\leq k<d$…

Let $X\subset \mathbb{P}^n$ be a totally real variety, let $P_X$ denote its cone of nonnegative forms, and let $\mathcal{F}$ be a face of $P_X$. Face-relative-interior conjecture.…

Let $n=2$ and let $T$ be any tensor defining the quartic-regularized model $m_3$. A necessary and sufficient global optimality condition with no relaxation gap should hold for ever…

Let $K_A$ and $K_B$ be state spaces, let $G$ be a game, and let $\beta^*(G)$ and $\gamma^*(G)$ denote the corresponding optimization values over assemblages and bipartite states wi…

Consider the globally rationally stable polynomial vector field … and the SOS tests for rational stability applied globally on $\Omega=\mathbb{R}^3$, with fixed gain bound $M=1.5$…

Thin-ray conjecture. The function $f$ is unbounded below on $\mathcal P\cap\mathbb Z^n$ if and only if there exists a ray $\operatorname{ray}(y,d)$ of $\mathcal P$ such that, for e…

Let a polynomial optimization problem have a sparse moment relaxation and the corresponding dense moment relaxation, with relaxation order measured by the parameter in the respecti…

The heuristics $\mathbf{H1}$ and $\mathbf{H2}$ are methods for strengthening moment relaxations, with $\mathbf{H1}$ depending more weakly on the quality of available upper bounds t…

Concave multilinearization inequality. Then

Linear approximation-error conjecture. The bound is

Let $p$ be a low-degree polynomial on the unit sphere. Sphere-optimization hardness conjecture. Approximately optimizing $p$ over the unit sphere is conjectured to be computational…

Generic finite convergence conjecture. There exists a finite set of polynomials $\varphi_1,\ldots,\varphi_L$ in the coefficients of $f$, $a$, $b$, and $c$ such that, if

Finite convergence conjecture. The hierarchy has finite convergence under the optimality condition assumptions.

Mordell's inequality conjecture. The maximum of $\Delta$ is $n^n$ for $n\le 5$; equivalently, the complex polynomial optimization problem

Let $X \subseteq \mathbb{C}^n$ be an irreducible affine variety, and let $\overline{X} \subset \mathbb{P}^n$ be its projective closure. Let $Q_{\infty}$ denote the quadric at infin…

Finite-adaptability tractability conjecture. These theorems can be extended to any $k$.

Generalized Lax conjecture. Every hyperbolic program is equivalent to a semidefinite program.

Slow-convergence conjecture. The convergence of the bounds as $\alpha$ increases is slow because, for degree-$\alpha$ polynomial $\varphi$, the expression for $\mathcal{L}\varphi$…

Consider the polynomial optimization problem in equation and its first-order relaxation in the Lasserre hierarchy. First-order relaxation exactness conjecture. Objective value exac…