Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane

We complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in $\mathbb{R}^2$. Previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral region in $\mathbb{R}^2$ can be done in polynomial time, while optimizing a quartic polynomial in the same type of region is NP-hard. We close the gap by showing that this problem can be solved in polynomial time for cubic polynomials. Furthermore, we show that the problem of minimizing a homogeneous polynomial of any fixed degree over the integer points in a bounded polyhedron in $\mathbb{R}^2$ is solvable in polynomial time. We show that this holds for polynomials that can be translated into homogeneous polynomials, even when the translation vector is unknown. We demonstrate that such problems in the unbounded case can have smallest optimal solutions of exponential size in the size of the input, thus requiring a compact representation of solutions for a general polynomial time algorithm for the unbounded case

SBV regularity for Hamilton-Jacobi equations in Rn\mathbb R^n

In this paper we study the regularity of viscosity solutions to the following Hamilton-Jacobi equations $$\partial_t u + H(D_{x} u)=0 \qquad \textrm{in} \Omega\subset \mathbb R\times \mathbb R^{n} .$$ In particular, under the assumption that the Hamiltonian $H\in C^2(\mathbb R^n)$ is uniformly convex, we prove that $D_{x}u$ and $\partial_t u$ belong to the class $SBV_{loc}(\Omega)$.Comment: 15 page

The Cartan-Hadamard conjecture and The Little Prince

The generalized Cartan-Hadamard conjecture says that if $\Omega$ is a domain with fixed volume in a complete, simply connected Riemannian $n$-manifold $M$ with sectional curvature $K \le \kappa \le 0$, then the boundary of $\Omega$ has the least possible boundary volume when $\Omega$ is a round $n$-ball with constant curvature $K=\kappa$. The case $n=2$ and $\kappa=0$ is an old result of Weil. We give a unified proof of this conjecture in dimensions $n=2$ and $n=4$ when $\kappa=0$, and a special case of the conjecture for \kappa \textless{} 0 and a version for \kappa \textgreater{} 0. Our argument uses a new interpretation, based on optical transport, optimal transport, and linear programming, of Croke's proof for $n=4$ and $\kappa=0$. The generalization to $n=4$ and $\kappa \ne 0$ is a new result. As Croke implicitly did, we relax the curvature condition $K \le \kappa$ to a weaker candle condition $Candle(\kappa)$ or $LCD(\kappa)$.We also find counterexamples to a na\"ive version of the Cartan-Hadamard conjecture: For every \varepsilon \textgreater{} 0, there is a Riemannian 3-ball $\Omega$ with $(1-\varepsilon)$-pinched negative curvature, and with boundary volume bounded by a function of $\varepsilon$ and with arbitrarily large volume.We begin with a pointwise isoperimetric problem called "the problem of the Little Prince." Its proof becomes part of the more general method.Comment: v3: significant rewritting of some proofs, a mistake in the proof of the ball counter-example has been correcte

A general and intuitive envelope theorem

We present an envelope theorem for establishing first-order conditions in decision problems involving continuous and discrete choices. Our theorem accommodates general dynamic programming problems, even with unbounded marginal utilities. And, unlike classical envelope theorems that focus only on differentiating value functions, we accommodate other endogenous functions such as default probabilities and interest rates. Our main technical ingredient is how we establish the differentiability of a function at a point: we sandwich the function between two differentiable functions from above and below. Our theory is widely applicable. In unsecured credit models, neither interest rates nor continuation values are globally differentiable. Nevertheless, we establish an Euler equation involving marginal prices and values. In adjustment cost models, we show that first-order conditions apply universally, even if optimal policies are not (S,s). Finally, we incorporate indivisible choices into a classic dynamic insurance analysis

A path following algorithm for the graph matching problem

We propose a convex-concave programming approach for the labeled weighted graph matching problem. The convex-concave programming formulation is obtained by rewriting the weighted graph matching problem as a least-square problem on the set of permutation matrices and relaxing it to two different optimization problems: a quadratic convex and a quadratic concave optimization problem on the set of doubly stochastic matrices. The concave relaxation has the same global minimum as the initial graph matching problem, but the search for its global minimum is also a hard combinatorial problem. We therefore construct an approximation of the concave problem solution by following a solution path of a convex-concave problem obtained by linear interpolation of the convex and concave formulations, starting from the convex relaxation. This method allows to easily integrate the information on graph label similarities into the optimization problem, and therefore to perform labeled weighted graph matching. The algorithm is compared with some of the best performing graph matching methods on four datasets: simulated graphs, QAPLib, retina vessel images and handwritten chinese characters. In all cases, the results are competitive with the state-of-the-art.Comment: 23 pages, 13 figures,typo correction, new results in sections 4,5,