arXiv subjects

Immanuel Bomze

Publications and source records attributed to Immanuel Bomze.

9 recordsLinked to original sources

arXiv2026

The Maximum Clique Problem under Adversarial Uncertainty: a min-max approach

We analyze the problem of identifying large cliques in graphs that are affected by adversarial uncertainty. More specifically, we consider a new formulation, namely the adversarial maximum clique problem, which extends the classical maximum-clique problem to graphs with edges strategically perturbed by an adversary. The proposed mathematical model is thus formulated as a two-player zero-sum game between a clique seeker and an opposing agent. Inspired by regularized continuous reformulations of the maximum-clique problem, we derive a penalized continuous formulation leading to a nonconvex and nonsmooth optimization problem. We further introduce the notion of stable global solutions, namely points remaining optimal under small perturbations of the penalty parameters, and prove an equivalence between stable global solutions of the continuous reformulation and largest cliques that are common to all the adversarially perturbed graphs. In order to solve the given nonsmooth problem, we develop a first-order and projection-free algorithm based on generalized subdifferential calculus in the sense of Clarke and Goldstein, and establish global sublinear convergence rates for it. Finally, we report numerical experiments on benchmark instances showing that the proposed method efficiently detects large common cliques.

math.OC↗

arXiv2024

Finding quadratic underestimators for optimal value functions of nonconvex all-quadratic problems via copositive optimization

Modeling parts of an optimization problem as an optimal value function that depends on a top-level decision variable is a regular occurrence in optimization and an essential ingredient for methods such as Benders Decomposition. It often allows for the disentanglement of computational complexity and exploitation of special structures in the lower-level problem that define the optimal value functions. If this problem is convex, duality theory can be used to build piecewise affine models of the optimal value function over which the top-level problem can be optimized efficiently. In this text, we are interested in the optimal value function of an all-quadratic problem (also called quadratically constrained quadratic problem, QCQP) which is not necessarily convex, so that duality theory can not be applied without introducing a generally unquantifiable relaxation error. This issue can be bypassed by employing copositive reformulations of the underlying QCQP. We investigate two ways to parametrize these by the top-level variable. The first one leads to a copositive characterization of an underestimator that is sandwiched between the convex envelope of the optimal value function and that envelope's lower-semicontinuous hull. The dual of that characterization allows us to derive affine underestimators. The second parametrization yields an alternative characterization of the optimal value function itself, which other than the original version has an exact dual counterpart. From the latter, we can derive convex and nonconvex quadratic underestimators of the optimal value function. In fact, we can show that any quadratic underestimator is associated with a dual feasible solution in a certain sense.

math.OC↗

arXiv2024

Tighter yet more tractable relaxations and nontrivial instance generation for sparse standard quadratic optimization

The Standard Quadratic optimization Problem (StQP), arguably the simplest among all classes of NP-hard optimization problems, consists of extremizing a quadratic form (the simplest nonlinear polynomial) over the standard simplex (the simplest polytope/compact feasible set). As a problem class, StQPs may be nonconvex with an exponential number of inefficient local solutions. StQPs arise in a multitude of applications, among them mathematical finance, machine learning (clustering), and modeling in biosciences (e.g., selection and ecology). This paper deals with such StQPs under an additional sparsity or cardinality constraint, which, even for convex objectives, renders NP-hard problems. One motivation to study StQPs under such sparsity restrictions is the high-dimensional portfolio selection problem with too many assets to handle, in particular, in the presence of transaction costs. Here, relying on modern conic optimization techniques, we present tractable convex relaxations for this relevant but difficult problem. We propose novel equivalent reformulations of these relaxations with significant dimensional reduction, which is essential for the tractability of these relaxations when the problem size grows. Moreover, we propose an instance generation procedure which systematically avoids too easy instances. Our extensive computational results illustrate the high quality of the relaxation bounds in a significant number of instances. Furthermore, in contrast with exact mixed-integer quadratic programming models, the solution time of the relaxations is very robust to the choices of the problem parameters. In particular, the reduced formulations achieve significant improvements in terms of the solution time over their counterparts.

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arXiv2024

Feature selection in linear SVMs via a hard cardinality constraint: a scalable SDP decomposition approach

In this paper, we study the embedded feature selection problem in linear Support Vector Machines (SVMs), in which a cardinality constraint is employed, leading to an interpretable classification model. The problem is NP-hard due to the presence of the cardinality constraint, even though the original linear SVM amounts to a problem solvable in polynomial time. To handle the hard problem, we first introduce two mixed-integer formulations for which novel semidefinite relaxations are proposed. Exploiting the sparsity pattern of the relaxations, we decompose the problems and obtain equivalent relaxations in a much smaller cone, making the conic approaches scalable. To make the best usage of the decomposed relaxations, we propose heuristics using the information of its optimal solution. Moreover, an exact procedure is proposed by solving a sequence of mixed-integer decomposed semidefinite optimization problems. Numerical results on classical benchmarking datasets are reported, showing the efficiency and effectiveness of our approach.

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arXiv2023

Projection free methods on product domains

Projection-free block-coordinate methods avoid high computational cost per iteration and at the same time exploit the particular problem structure of product domains. Frank-Wolfe-like approaches rank among the most popular ones of this type. However, as observed in the literature, there was a gap between the classical Frank-Wolfe theory and the block-coordinate case. Moreover, most of previous research concentrated on convex objectives. This study now deals also with the non-convex case and reduces above-mentioned theory gap, in combining a new, fully developed convergence theory with novel active set identification results which ensure that inherent sparsity of solutions can be exploited in an efficient way. Preliminary numerical experiments seem to justify our approach and also show promising results for obtaining global solutions in the non-convex case.

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arXiv2023

On Tractable Convex Relaxations of Standard Quadratic Optimization Problems under Sparsity Constraints

Standard quadratic optimization problems (StQPs) provide a versatile modelling tool in various applications. In this paper, we consider StQPs with a hard sparsity constraint, referred to as sparse StQPs. We focus on various tractable convex relaxations of sparse StQPs arising from a mixed-binary quadratic formulation, namely, the linear optimization relaxation given by the reformulation-linearization technique, the Shor relaxation, and the relaxation resulting from their combination. We establish several structural properties of these relaxations in relation to the corresponding relaxations of StQPs without any sparsity constraints, and pay particular attention to the rank-one feasible solutions retained by these relaxations. We then utilize these relations to establish several results about the quality of the lower bounds arising from different relaxations. We also present several conditions that ensure the exactness of each relaxation.

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arXiv2023

The star formation history of the Sco-Cen association. Coherent star formation patterns in space and time

We reconstruct the star formation history of the Sco-Cen OB association using a novel high-resolution age map of the region. We develop an approach to produce robust ages for Sco-Cen's recently identified 37 stellar clusters using the SigMA algorithm. The Sco-Cen star formation timeline reveals four periods of enhanced star formation activity, or bursts, remarkably separated by about 5 Myr. Of these, the second burst, which occurred about 15 million years ago, is by far the dominant, and most of Sco-Cen's stars and clusters were in place by the end of this burst. The formation of stars and clusters in Sco-Cen is correlated but not linearly, implying that more stars were formed per cluster during the peak of the star formation rate. Most of the clusters that are large enough to have supernova precursors were formed during the 15 Myr period. Star and cluster formation activity has been continuously declining since then. We have clear evidence that Sco-Cen formed from the inside out and contains 100-pc long chains of contiguous clusters exhibiting well-defined age gradients, from massive older clusters to smaller young clusters. These observables suggest an important role for feedback in forming about half of Sco-Cen stars, although follow-up work is needed to quantify this statement. Finally, we confirm that the Upper-Sco age controversy discussed in the literature during the last decades is solved: the nine clusters previously lumped together as Upper-Sco, a benchmark region for planet formation studies, exhibit a wide range of ages from 3 to 19 Myr.

astro-ph.SR↗

arXiv2022

Significance Mode Analysis (SigMA) for hierarchical structures. An application to the Sco-Cen OB association

We present a new clustering method, Significance Mode Analysis (SigMA), to extract co-spatial and co-moving stellar populations from large-scale surveys such as ESA Gaia. The method studies the topological properties of the density field in the multidimensional phase space. We validate SigMA on simulated clusters and find that it outperforms competing methods, especially in cases where many clusters are closely spaced. We apply the new method to Gaia DR3 data of the closest OB association to Earth, Scorpio-Centaurus (Sco-Cen), and find more than 13,000 co-moving young objects, with about 19% of these having a sub-stellar mass. SigMA finds 37 co-moving clusters in Sco-Cen. These clusters are independently validated by their narrow HRD sequences and, to a certain extent, by their association with massive stars too bright for Gaia, hence unknown to SigMA. We compare our results with similar recent work and find that the SigMA algorithm recovers richer populations, is able to distinguish clusters with velocity differences down to about 0.5 km s$^{-1}$, and reaches cluster volume densities as low as 0.01 sources/pc$^3$. The 3D distribution of these 37 coeval clusters implies a larger extent and volume for the Sco-Cen OB association than typically assumed in the literature. Additionally, we find the association to be more actively star-forming and dynamically more complex than previously thought. We confirm that the star-forming molecular clouds in the Sco-Cen region, namely, Ophiuchus, L134/L183, Pipe Nebula, Corona Australis, Lupus, and Chamaeleon, are part of the Sco-Cen The application of SigMA to Sco-Cen demonstrates that advanced machine learning tools applied to the superb Gaia data allows to construct an accurate census of the young populations, to quantify their dynamics, and to reconstruct the recent star formation history of the local Milky Way.

astro-ph.GA↗

arXiv2022

Uncertainty Preferences in Robust Mixed-Integer Linear Optimization with Endogenous Uncertainty

In robust optimization one seeks to make a decision under uncertainty, where the goal is to find the solution with the best worst-case performance. The set of possible realizations of the uncertain data is described by a so-called uncertainty set. In many scenarios, a decision maker may influence the uncertainty regime she is facing, for example, by investing in market research, or in machines which work with higher precision. Recently, this situation was addressed in the literature by introducing decision dependent uncertainty sets (endogenous uncertainty), i.e., uncertainty sets whose structure depends on (typically discrete) decision variables. In this way, one can model the trade-off between reducing the cost of robustness versus the cost of the investment necessary for influencing the uncertainty. However, there is another trade-off to be made here. With different uncertainty regimes, not only do the worst-case optimal solutions vary, but also other aspects of that solutions such as max-regret, best-case performance or predictability of the performance. A decision maker may still be interested in having a performance guarantee, but at the same time be willing to forgo superior worst-case performance if those other aspects can be enhanced by switching to a suitable uncertainty regime. We introduce the notion of uncertainty preference in order to capture such stances. We present three ways to formalize uncertainty preferences and study the resulting mathematical models. The goal is to have reformulations/approximations of these models which can be solved with standard methods. The workhorse is mixed-integer linear and conic optimization. We apply our framework to the uncertain shortest path problem and conduct numerical experiments for the resulting models. We can demonstrate that our models can be handled very well by standard mixed-integer linear solvers.

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