Related papers: Self-concordance is NP-hard
We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list here includes: determining the feasibility of a system of bilinear equations, deciding whether a 3-tensor…
Functions that are piecewise defined are a common sight in mathematics while convexity is a property especially desired in optimization. Suppose now a piecewise-defined function is convex on each of its defining components - when can we…
We prove that persuasion is an NP-complete problem.
Self-concordance is the most important property required for barriers in convex programming. It is intrinsically linked to the affine structure of the underlying space. Here we introduce an alternative notion of self-concordance which is…
For convex univalent functions we give instances where the sharp bound for various coefficient functionals are identical to those for the corresponding bound for the inverse function. We give instances where the sharp bounds differ and also…
In this work, we discuss the continuity of $h$-convex functions by introducing the concepts of $h$-convex curves ($h$-cord). Geometric interpretation of $h$-convexity is given. The fact that for a $h$-continuous function $f$, is being…
The self-concordant-like property of a smooth convex function is a new analytical structure that generalizes the self-concordant notion. While a wide variety of important applications feature the self-concordant-like property, this concept…
We show that approximating the trace norm contraction coefficient of a quantum channel within a constant factor is NP-hard. Equivalently, this shows that determining the optimal success probability for encoding a bit in a quantum system…
We investigate determining the exact bounds of the frequencies of conjunctions based on frequent sets. Our scenario is an important special case of some general probabilistic logic problems that are known to be intractable. We show that…
We show that the deviation between the slopes of two convex functions controls the deviation between the functions themselves. This result reveals that the slope -- a one dimensional construct -- robustly determines convex functions, up to…
We prove that, in general, given a $p$-harmonic map $F:M\to N$ and a convex function $H:N\to\mathbb{R}$, the composition $H\circ F$ is not $p$-subharmonic. By assuming some rotational symmetry on manifolds and functions, we reduce the…
The convex feasibility problem (CFP) is to find a feasible point in the intersection of finitely many convex and closed sets. If the intersection is empty then the CFP is inconsistent and a feasible point does not exist. However,…
We extend the standard notion of self-concordance to non-convex optimization and develop a family of second-order algorithms with global convergence guarantees. In particular, two function classes -- \textit{weakly self-concordant}…
In this note we prove NP-hardness of the following problem: Given a set of matrices, is there a convex combination of those that is a nonsingular M-matrix? Via known characterizations of M-matrices, our result establishes NP-hardness of…
Proving that a finitely generated convex cone is closed is often considered the most difficult part of geometric proofs of Farkas' lemma. We provide a short simple proof of this fact and (for completeness) derive Farkas' lemma from it using…
We consider the class of convex minimization problems, composed of a self-concordant function, such as the $\log\det$ metric, a convex data fidelity term $h(\cdot)$ and, a regularizing -- possibly non-smooth -- function $g(\cdot)$. This…
In this paper, we prove the problem of stabbing a set of disjoint bends by a convex stabber to be NP-hard. We also consider the optimization version of the convex stabber problem and prove this problem to be APX-hard for sets of line…
Barrier methods play a central role in the theory and practice of convex optimization. One of the most general and successful analyses of barrier methods for convex optimization, due to Nesterov and Nemirovskii, relies on the notion of…
0-1 Knapsack is a fundamental NP-complete problem. In this article we prove that it remains NP-complete even when the weights of the objects in the packing constraints and their values in the objective function satisfy specific stringent…
We show that the problem of finding a Resolution refutation that is at most polynomially longer than a shortest one is NP-hard. In the parlance of proof complexity, Resolution is not automatizable unless P = NP. Indeed, we show it is…