About

Pavlo Krokhmal is a Professor of Systems and Industrial Engineering and the Director of Industrial Engineering at the University of Arizona. He is a member of the Graduate Faculty and has a background in operations research, mechanics of solids, and applied mathematics, holding PhDs from the University of Florida and Kyiv National Taras Shevchenko University. His research interests encompass stochastic optimization, decision making under uncertainty, risk analysis, probabilistic and stochastic combinatorial optimization, financial engineering, optimal trading strategies, and the pricing of derivatives. He has contributed to the fields of multidisciplinary optimization, cooperative control and decision making, computational and applied mathematics, and continuum mechanics. Krokhmal has an extensive publication record, including book chapters and journal articles, and has been recognized with numerous awards and fellowships for his research and teaching excellence.

Research topics

• Computer Science
• Mathematical optimization
• Mathematics
• Algorithm
• Artificial Intelligence
• Engineering
• Physics
• Statistics
• Discrete mathematics
• Combinatorics
• Electrical engineering
• Computer vision

Selected publications

• PDE-constrained optimization for vibration mitigation in active structures

  Mechanics of Advanced Materials and Structures · 2026-04-24

  articleSenior author
  PublisherDOI

• Preface: global optimization algorithms and applications

  Computational Optimization and Applications · 2025-10-03

  articleOpen access
  PublisherOA PDFDOI

• Finding the Maximum Subgraph of Prescribed Strength

  Networks · 2025-12-10

  articleSenior author

  ABSTRACT The strength of a graph was introduced by Cunningham (1985) as the minimum ratio of the number of edges that could be removed from the graph to the number (minus one) of the connected components created by such a removal. The role of the strength of a graph as a connectivity and resilience measure is further highlighted by its connection to the spanning tree packing number of a graph, namely, the maximum number of edge‐disjoint spanning trees that can be contained in a graph. In this work, we investigate the question of determining a subset of a given graph's vertices of maximum cardinality that induces a subgraph of at least the prescribed strength. We show that the stated problem is polynomially solvable, and present the corresponding algorithm. In addition, we propose a mathematical programming‐based cutting plane method for computing graph strength, which can be integrated into larger mathematical programming models. Numerical experiments on a diverse array of real‐world graphs illustrate the computational properties of the proposed algorithms.

  PublisherDOI

• Comparisons of Unsupervised and Supervised Machine Learning Algorithms for Prediction of Damage in Composites From 3D Micro-CT Images

  2025-05-05

  article

  Abstract The objective of this work was to conduct statistically meaningful comparisons of the unsupervised and supervised machine learning (ML) methods for damage segmentation in composites from 3D Micro Computed Tomography (micro-CT) data and explore synergy between unsupervised and supervised ML. This synergy enables one to combine strong mathematical rigor of the unsupervised ML methods with flexibility and accessibility of the supervised ML algorithms. The unsupervised ML method relied on the statistical distances in conjunction with grayscale threshold intensity segmentation to isolate damage present in high resolution image data. The deep learning models used in this work were based on the U-Net and FC-DenseNet architectures. Both unsupervised and supervised ML methods were applied to the analysis of low velocity impact damage in the carbon fiber reinforced polymer (CFRP) composites. The performance of the methods was assessed using metrics from the statistical classification theory.

  PublisherDOI

• Chance-Constrained Set Multicover Problem

  arXiv (Cornell University) · 2024-11-06

  preprintOpen accessSenior author

  We consider a variant of the set covering problem with uncertain parameters, which we refer to as the chance-constrained set multicover problem (CC-SMCP). In this problem, we assume that there is uncertainty regarding whether a selected set can cover an item, and the objective is to determine a minimum-cost combination of sets that covers each item $i$ at least $k_i$ times with a prescribed probability. To tackle CC-SMCP, we employ techniques of enumerative combinatorics, discrete probability distributions, and combinatorial optimization to derive exact equivalent deterministic reformulations that feature a hierarchy of bounds, and develop the corresponding outer-approximation (OA) algorithm. Additionally, we consider reducing the number of chance constraints via vector dominance relations and reformulate two special cases of CC-SMCP using the ``log-transformation" method and binomial distribution properties. Theoretical results on sampling-based methods, i.e., the sample average approximation (SAA) method and the importance sampling (IS) method, are also studied to approximate the true optimal value of CC-SMCP under a finite discrete probability space. Our numerical experiments demonstrate the effectiveness of the proposed OA method, particularly in scenarios with sparse probability matrices, outperforming sampling-based approaches in most cases and validating the practical applicability of our solution approaches.

  PublisherOA PDFDOI

• Unsupervised Machine Learning for Automatic Image Segmentation of Impact Damage in CFRP Composites

  Applied Composite Materials · 2024-07-13 · 6 citations

  articleSenior author
  PublisherDOI

• Risk‐averse optimization and resilient network flows

  Networks · 2023-05-13 · 2 citations

  articleSenior authorCorresponding

  Abstract We propose an approach to constructing metrics of network resilience, where resilience is understood as the network's amenability to restoring its optimal or near‐optimal operations subsequent to unforeseen (stochastic) disruptions of its topology or operational parameters, and illustrated it on the examples of the resilient maximum network flow problem and the resilient minimum cost network problem. Specifically, the network flows in these problems are designed for resilience against unpredictable losses of network carrying capacity, and the mechanism of attaining a degree of resilience is through preallocation of resources toward (at least partial) restoration of the capacities of the arcs. The obtained formulations of resilient network flow problems possess a number of useful properties, for example, similarly to the standard network flow problems, the network flow is integral if the arc capacities, costs, and so forth, are integral. It is also shown that the proposed formulations of resilient network flow problems can be viewed as “network measures of risk”, similar in properties and behavior to convex measures of risk. Efficient decomposition algorithms have been proposed for both the resilient maximum network flow problem and the resilient minimum cost network flow problem, and a study of the network flow resilience as a function of network's structure has been conducted on networks with three types of topology: that of uniform random graphs, scale‐free graphs, and grid graphs.

  PublisherDOI

• Asymptotic bounds for clustering problems in random graphs

  Networks · 2023-12-13 · 1 citations

  articleOpen accessSenior author

  Abstract Graph clustering is an important problem in network analysis. This problem can be approached by first finding a large cluster subgraph (i.e., a subgraph in which every connected component is a complete graph), perhaps in a relaxed form (connected components may have missing edges), and then assigning each of the remaining vertices to one of the connected components of the cluster subgraph according to some optimization criteria. The more vertices can be included in the initial cluster subgraph (also referred to as independent union of clusters), the more “clusterable” the graph is. This paper proposes a framework for establishing asymptotic bounds on the cardinality of independent unions of clusters in Erdős‐Rényi random graphs with constant , referred to as uniform random graphs. In particular, sufficient conditions ensuring (where is the number of nodes) upper bounds with probability 1 are developed and shown to be applicable for the maximum independent union of cliques as well as some clique relaxations. In addition, it is shown that every graph must have an independent union of cliques of cardinality at least . Since this bound is asymptotically tight on uniform random graphs, this suggests that these graphs can be viewed as a “least clusterable” class of graphs.

  PublisherOA PDFDOI

• Stochastic and Risk Averse Maximum Subgraph Problems

  Encyclopedia of Optimization · 2022-09-21

  book-chapterSenior author
  PublisherDOI

• Unsupervised Machine Learning Algorithms for Analysis of Low Velocity Impact Damage in Composite Structures From CT Image Data

  2022 · 2 citations

  Senior authorCorresponding
  • Artificial Intelligence
  • Computer Science
  • Artificial Intelligence

  Abstract In this work, novel unsupervised machine learning (ML) algorithms for automatic image segmentation for the analysis of the micro-CT data for impact damage assessment in the composite materials have been developed. The algorithms are based on the statistical distances including the Kullback-Leibler divergence, the Helling distance, and the Renyi divergence. The developed algorithms have been applied to the analysis of low velocity impact damage in carbon fiber reinforced polymer (CFRP) composites. The grayscale images from the CT scans of the impacted CFRP specimens have been analyzed to identify and isolate impact damage and optimal statistics-based damage thresholds have been found. The results show that the developed algorithms enable not only an automatic image segmentation, but also deliver statistics-based rigorous damage thresholds.

  PublisherDOI

Frequent coauthors

• Maciej Rysz

  Miami University

  13 shared

• Stan Uryasev

  10 shared

• Olesya I. Zhupanska

  9 shared

• Robert Murphey

  9 shared

• Eduardo L. Pasiliao

  United States Air Force Research Laboratory

  9 shared

• Alexander Vinel

  Auburn University

  7 shared

• Pãnos M. Pardalos

  University of Florida

  6 shared

• David E. Jeffcoat

  University of Florida

  6 shared

Awards & honors

• Diploma in the Competition of Young Scientists and Students…
• Soros Student Award International Soros Science and Educatio…
• Scholarship for scientific and academic achievements Nationa…
• Air Force Summer Faculty Fellowship Award Air Force Office o…
• Air Force Office of Scientific Research, Summer I 2018