Running Intersection Research Articles

Classical iterative proportional scaling of log-linear models with rational maximum likelihood estimator

In this work we investigate multipartition models, the subset of log-linear models for which one can perform the classical iterative proportional scaling (IPS) algorithm to numerically compute the maximum likelihood estimate (MLE). Multipartition models include families of models such as hierarchical models and balanced, stratified staged trees. We define a sufficient condition, called the Generalized Running Intersection Property (GRIP), on the matrix representation of a multipartition model under which the classical IPS algorithm produces the exact MLE in one cycle. In this case, the MLE is a rational function of the data. Additionally we connect the GRIP to the toric fiber product and to previous results for hierarchical models and balanced, stratified staged trees. This leads to a characterization of balanced, stratified staged trees in terms of the GRIP.

A Sparse Version of Reznick’s Positivstellensatz

If f is a positive definite form, Reznick’s Positivstellensatz states that there exists [Formula: see text] such that [Formula: see text] is a sum of squares of polynomials. Assuming that f can be written as a sum of forms [Formula: see text], where each fl depends on a subset of the initial variables, and assuming that these subsets satisfy the so-called running intersection property, we provide a sparse version of Reznick’s Positivstellensatz. Namely, there exists [Formula: see text] such that [Formula: see text], where σl is a sum of squares of polynomials, Hl is a uniform polynomial denominator, and both polynomials [Formula: see text] involve the same variables as fl for each [Formula: see text]. In other words, the sparsity pattern of f is also reflected in this sparse version of Reznick’s certificate of positivity. We next use this result to also obtain positivity certificates for (i) polynomials nonnegative on the whole space and (ii) polynomials nonnegative on a (possibly noncompact) basic semialgebraic set, assuming that the input data satisfy the running intersection property. Both are sparse versions of a positivity certificate from Putinar and Vasilescu. Funding: V. Magron was supported by the Fondation Mathématique Jacques Hadamard Programme Gaspard Monge for Optimization (Exact Polynomial Optimization with Innovative Certifed Schemes project). This work has benefited from the Tremplin European Research Council Starting [Grant ANR-18-ERC2-0004-01] (Tremplin-Certified Optimization for Cyber-Physical Systems project), the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Actions [Grant 813211] (Polynomial Optimization, Efficiency through Moments and Algebra) as well as from the Artificial Intelligence Interdisciplinary Institute Artificial and Natural Intelligence Toulouse Institute funding, through the French “Investing for the Future Programme d’Investissements d’Avenir3” program [Grant n°ANR-19-PI3A-0004].

The Running Intersection Relaxation of the Multilinear Polytope

The multilinear polytope of a hypergraph is the convex hull of a set of binary points satisfying a collection of multilinear equations. We introduce the running intersection inequalities, a new class of facet-defining inequalities for the multilinear polytope. Accordingly, we define a new polyhedral relaxation of the multilinear polytope, referred to as the running intersection relaxation, and identify conditions under which this relaxation is tight. Namely, we show that for kite-free beta-acyclic hypergraphs, a class that lies between gamma-acyclic and beta-acyclic hypergraphs, the running intersection relaxation coincides with the multilinear polytope and it admits a polynomial size extended formulation.

Compatibility, desirability, and the running intersection property

Compatibility is the problem of checking whether some given probabilistic assessments have a common joint probabilistic model. When the assessments are unconditional, the problem is well established in the literature and finds a solution through the running intersection property (RIP). This is not the case of conditional assessments. In this paper, we study the compatibility problem in a very general setting: any possibility space, unrestricted domains, imprecise (and possibly degenerate) probabilities. We extend the unconditional case to our setting, thus generalising most of previous results in the literature. The conditional case turns out to be fundamentally different from the unconditional one. For such a case, we prove that the problem can still be solved in general by RIP but in a more involved way: by constructing a junction tree and propagating information over it. Still, RIP does not allow us to optimally take advantage of sparsity: in fact, conditional compatibility can be simplified further by joining junction trees with coherence graphs.

On the impact of running intersection inequalities for globally solving polynomial optimization problems

We consider global optimization of nonconvex problems whose factorable reformulations contain a collection of multilinear equations of the form $$z_e = \prod _{v \in e} {z_v}$$, $$e \in E$$, where E denotes a set of subsets of cardinality at least two of a ground set. Important special cases include multilinear and polynomial optimization problems. The multilinear polytope is the convex hull of the set of binary points z satisfying the system of multilinear equations given above. Recently Del Pia and Khajavirad introduced running intersection inequalities, a family of facet-defining inequalities for the multilinear polytope. In this paper we address the separation problem for this class of inequalities. We first prove that separating flower inequalities, a subclass of running intersection inequalities, is NP-hard. Subsequently, for multilinear polytopes of fixed degree, we devise an efficient polynomial-time algorithm for separating running intersection inequalities and embed the proposed cutting-plane generation scheme at every node of the branch-and-reduce global solver BARON. To evaluate the effectiveness of the proposed method we consider two test sets: randomly generated multilinear and polynomial optimization problems of degree three and four, and computer vision instances from an image restoration problem Results show that running intersection cuts significantly improve the performance of BARON and lead to an average CPU time reduction of 50% for the random test set and of 63% for the image restoration test set.

Robustness to Dependency in Portfolio Optimization Using Overlapping Marginals

In this paper, we develop a distributionally robust portfolio optimization model where the robustness is across different dependency structures among the random losses. For a Fréchet class of discrete distributions with overlapping marginals, we show that the distributionally robust portfolio optimization problem is efficiently solvable with linear programming. To guarantee the existence of a joint multivariate distribution consistent with the overlapping marginal information, we make use of a graph theoretic property known as the running intersection property. Building on this property, we develop a tight linear programming formulation to find the optimal portfolio that minimizes the worst-case conditional value-at-risk measure. Lastly, we use a data-driven approach with financial return data to identify the Fréchet class of distributions satisfying the running intersection property and then optimize the portfolio over this class of distributions. Numerical results in two different data sets show that the distributionally robust portfolio optimization model improves on the sample-based approach.

An efficient representation of chordal graphs

In this paper, a representation for chordal graphs called the compact representation, based on the running intersection property, is presented. It provides the means to immediately deduce several structural properties of a chordal graph such as a perfect elimination ordering, the minimal vertex separators and a clique-tree. These properties support an efficient algorithm for the construction of the compact representation. Simple characterizations of some subclasses of chordal graphs can be obtained using this representation.

Convergent SDP‐Relaxations in Polynomial Optimization with Sparsity

We consider a polynomial programming problem $\P$ on a compact basic semialgebraic set $\K\subset\R^n$, described by $m$ polynomial inequalities $g_j(X)\geq0$, and with criterion $f\in\R[X]$. We propose a hierarchy of semidefinite relaxations in the spirit of those of Waki e [SIAM J. Optim., 17 (2006), pp. 218-242]. In particular, the SDP-relaxation of order $r$ has the following two features: (a) The number of variables is $O(\kappa^{2r})$, where $\kappa=\max[\kappa_1,\kappa_2]$ with $\kappa_1$ (resp., $\kappa_2$) being the maximum number of variables appearing in the monomials of $f$ (resp., appearing in a single constraint $g_j(X)\geq0$). (b) The largest size of the linear matrix inequalities (LMIs) is $O(\kappa^r)$. This is to compare with the respective number of variables $O(n^{2r})$ and LMI size $O(n^r)$ in the original SDP-relaxations defined in [J. B. Lasserre, SIAM J. Optim., 11 (2001), pp. 796-817]. Therefore, great computational savings are expected in case of sparsity in the data $\{g_j,f\}$, i.e., when k is small, a frequent case in practical applications of interest. The novelty with respect to [H. Waki, S. Kim, M. Kojima, and M. Maramatsu, SIAM J. Optim., 17 (2006), pp. 218-242] is that we prove convergence to the global optimum of $\P$ when the sparsity pattern satisfies a condition often encountered in large size problems of practical applications, and known as the running intersection property in graph theory. In such cases, and as a by-product, we also obtain a new representation result for polynomials positive on a compact basic semialgebraic set, a sparse version of Putinar’s Positivstellensatz [M. Putinar, Indiana Univ. Math. J., 42 (1993), pp. 969-984].

Conditional independence and natural conditional functions

The concept of conditional independence (CI) within the framework of natural conditional functions (NCFs) is studied. An NCF is a function asribing natural numbers to possible states of the world; it is the central concept of Spohn's theory of deterministic epistemology. Basic properties of CI within this framework are recalled, and further results analogous to the results concerning probabilistic CI are proved. Firstly, the intersection of two CI-models is shown to be a CI-model. Using this, it is proved that CI-models for NCFs have no finite complete axiomatic characterization (by means of a simple deductive system describing relationships among CI-statements). The last part is devoted to the marginal problem for NCFs. It is shown that (pairwise) consonancy is equivalent to consistency iff the running intersection property holds.