In this paper, we consider the problem of finding bounds on the prices of options depending on multiple assets without assuming any underlying model on the price dynamics but only the absence of arbitrage opportunities. We formulate this as a generalized moment problem and utilize the well-known moment-sum-of-squares hierarchy of Lasserre to obtain bounds on the range of the possible prices. A complementary approach (also from Lasserre) is employed for comparison. We present several numerical examples to demonstrate the viability of our approach. The framework we consider makes it possible to incorporate different kinds of observable data, such as moment information, as well as observable prices of options on the assets of interest. History: Accepted by Antonio Frangioni, area editor for Design & Analysis of Algorithms–Continuous. Funding: This work was supported by the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement [Grant 813211 (POEMA)]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplementary Information [ https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.1220 ] or is available from the IJOC GitHub software repository ( https://github.com/INFORMSJoC ) at [ http://dx.doi.org/10.5281/zenodo.6602361 ].

We propose an analytic center cutting plane method to determine whether a matrix is completely positive and return a cut that separates it from the completely positive cone if not. This was stated as an open (computational) problem by Berman et al. [Berman A, Dur M, Shaked-Monderer N (2015) Open problems in the theory of completely positive and copositive matrices. Electronic J. Linear Algebra 29(1):46–58]. Our method optimizes over the intersection of a ball and the copositive cone, where membership is determined by solving a mixed-integer linear program suggested by Xia et al. [Xia W, Vera JC, Zuluaga LF (2020) Globally solving nonconvex quadratic programs via linear integer programming techniques. INFORMS J. Comput. 32(1):40–56]. Thus, our algorithm can, more generally, be used to solve any copositive optimization problem, provided one knows the radius of a ball containing an optimal solution. Numerical experiments show that the number of oracle calls (matrix copositivity checks) for our implementation scales well with the matrix size, growing roughly like [Formula: see text] for d × d matrices. The method is implemented in Julia and available at https://github.com/rileybadenbroek/CopositiveAnalyticCenter.jl . Summary of Contribution: Completely positive matrices play an important role in operations research. They allow many NP-hard problems to be formulated as optimization problems over a proper cone, which enables them to benefit from the duality theory of convex programming. We propose an analytic center cutting plane method to determine whether a matrix is completely positive by solving an optimization problem over the copositive cone. In fact, we can use our method to solve any copositive optimization problem, provided we know the radius of a ball containing an optimal solution. We emphasize numerical performance and stability in developing this method. A software implementation in Julia is provided.

Recent progress in solving quadratic assignment problems (QAPs) from the QAPLIB (Quadratic Assignment Problem Library) test set has come from mixed-integer linear or quadratic programming models that are solved in a branch-and-bound framework. Semidefinite programming (SDP) bounds for QAPs have also been studied in some detail, but their computational impact has been limited so far, mostly because of the restrictive size of the early relaxations. Some recent progress has been made by studying smaller SDP relaxations and by exploiting group symmetry in the QAP data. In this work, we introduce a new SDP relaxation, where the matrix variables are only of the order of the QAP dimension, and we show how one may exploit group symmetry in the problem data for this relaxation. We also provide a detailed numerical comparison with related bounds from the literature. In particular, we compute the best-known lower bounds for two QAPLIB instances.

A book with $k$ pages consists of a straight line (the spine) and $k$ half-planes (the pages), such that the boundary of each page is the spine. If a graph is drawn on a book with $k$ pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is a k-page book drawing (or simply a $k$-page drawing). The $k$-page crossing number $\nu_k(G)$ of a graph $G$ is the minimum number of crossings in a $k$-page drawing of $G$. In this paper we investigate the $k$-page crossing numbers of complete graphs. We use semidefinite programming techniques to give improved lower bounds on $\nu_k(K_n)$ for various values of $k$. We also use a maximum satisfiability reformulation to obtain a computer-aided calculation of the exact value of $\nu_k(K_n)$ for several values of $k$ and $n$. Finally, we investigate the best construction known for drawing $K_n$ in $k$ pages, calculate the resulting number of crossings, and discuss this upper bound in light of the new results reported in this paper.

The problem of minimizing a quadratic form over the standard simplex is known as the standard quadratic optimization problem (SQO). It is NP-hard, and contains the maximum stable set problem in graphs as a special case. In this note, we show that the SQO problem may be reformulated as an (exponentially sized) linear program (LP). This reformulation also suggests a hierarchy of polynomial-time solvable LP’s whose optimal values converge finitely to the optimal value of the SQO problem. The hierarchies of LP relaxations from the literature do not share this finite convergence property for SQO, and we review the relevant counterexamples.

It has been long conjectured that the crossing number $\Cr(K_{m,n})$ of the complete bipartite graph $K_{m,n}$ equals the Zarankiewicz number $Z(m,n):= \floor{\frac{m-1}{2}} \floor{\frac{m}{2}} \floor{\frac{n-1}{2}} \floor{\frac{n}{2}}$. Another longstanding conjecture states that the crossing number $\Cr(K_n)$ of the complete graph $K_n$ equals $Z(n):=\frac{1}{4}\smallfloor{\frac{n}{2}} \smallfloor{\frac{n-1}{2}} \smallfloor{\frac{n-2}{2}}\smallfloor{\frac{n-3}{2}}$. In this paper we show the following improved bounds on the asymptotic ratios of these crossing numbers and their conjectured values: \begin{itemize} \item[(i)] for each fixed $m\ge 9$, $\lim_{n\to\infty} \Cr(K_{m,n})/Z(m,n) \ge 0.83m/(m-1)$; \item[(ii)] $\lim_{n\to\infty} \Cr(K_{n,n})/Z(n,n) \ge 0.83$; and \item[(iii)] $\lim_{n\to\infty} \Cr(K_{n})/Z(n) \ge 0.83$. \end{itemize} The previous best known lower bounds were $0.8m/(m-1), 0.8$, and $0.8$, respectively. These improved bounds are obtained as a consequence of the new bound $\Cr(\ksn) \ge 2.1796n^2 - 4.5n$. To obtain this improved lower bound for $\Cr(\ksn)$, we use some elementary topological facts on drawings of $K_{2,7}$ to set up a quadratic program on $6!$ variables whose minimum p satisfies $\Cr(\ksn) \ge (p/2)n^2 - 4.5n$, and then use state-of-the-art quadratic optimization techniques combined with a bit of invariant theory of permutation groups to show that $p \ge 4.3593$.