CVaR Optimization and the Risk Quadrangle: the BackstoryThis talk takes advantage of the occasion of celebrating Stan Uryasev's 70th to tell the story of our joint research. The aim is to explain how it took off with the VaR-CVaR formula, moved to extending classical finance theory to generalized measures of deviation, and how that led to the connections with statistics embodied in the risk quadrangle. Broad mathematical theories never emerge all at once, but evolve as new ideas emerge over time through insightful discussions and interactions, as will be illustrated.

Reinforcement Learning with Markov Risk Measures and Multipattern Risk ApproximationFor a risk-averse finite-horizon Markov Decision Problem, we introduce a special class of Markov coherent risk measures, called mini-batch measures. We also define the class of multipattern risk-averse problems that generalizes the class of linear systems. We use both concepts in a feature-based Q-learning method with multipattern Q-factor approximation, and we prove a high-probability regret bound. The theoretical results are illustrated on a stochastic assignment problem and a short-horizon multi-armed bandit problem.

Robustness Measures for Distributionally Robust Portfolio Optimization under Gaussian Mixture DistributionIn this talk, we will examine a distributionally robust optimization (DRO) model for portfolio selection that maximizes exponential expected utility under a Gaussian mixture model. Since Gaussian mixture models involve parameters at different hierarchical levels—specifically, the mixture weight vector and the mean vectors and covariance matrices of the individual normal distributions—it is necessary to consider two distinct types of uncertainty sets. We address cases where these uncertainty sets are defined based on either Kullback–Leibler divergence or optimal transport distance; we demonstrate that, in both instances, the problem can be reformulated as a convex cone optimization problem and solved using existing solvers such as MOSEK. Furthermore, we interpret the regularization term derived from the DRO objective function as a measure of robustness, derive a formula for worst-case sensitivity (WCS) applicable to these two types of uncertainty, and analyze the implications of choosing different uncertainty sets.

Planogram: A Multi-dimensional Physical Location Planning System for DC NetworksThe unprecedented demand for hyperscale data centers requires operators to scale physical footprints rapidly while managing complex resource dependencies. Traditional, manual site layout and topology mapping processes cannot scale to meet these timelines. To address this, we built an automated end-to-end planning pipeline featuring Planogram, a novel multi-dimensional optimization solution at its core. Powered by a lexicographic-minimum Mixed-Integer Linear Programming (lex-min MILP) formulation, Planogram systematically respects strict physical constraints—such as cable lengths, power, and cooling boundaries—while optimizing multi-tiered design objectives to map the logical network graph directly onto the physical floor plan. Integrating Planogram into the design pipeline significantly accelerates deployment velocity and guarantees consistency.

Machine Learning in “Human-Hard” Problems of Image Recognition and AnalysisIn this talk we discuss Machine Learning (ML) approaches to image recognition and analysis in engineering applications where the corresponding image processing tasks represent a challenge for humans. We demonstrate how unsupervised and supervised ML models based on relatively simple optimization formulations are capable of yielding results superior to those produced by significantly more computationally intensive mainstream approaches, such as convolutional neural networks (CNNs). In the first part of the talk, the problem of image segmentation of low velocity impact damage in composite materials is considered. The developed unsupervised ML approach is based on the use of non-parametric statistical models in conjunction with the so-called intensity-based segmentation, enabling one to isolate the damage regions in the grayscale micro computed tomography (micro-CT) scans of the impacted composite plates. The proposed method involves minimization of f-divergence metrics, such as the Kullback–Leibler divergence, the Hellinger distance, and the Renyi divergence, and is demonstrated to yield results superior to other unsupervised and supervised ML techniques, such as CNNs. In the second part, we consider the problem of target recognition and identification in Synthetic Aperture Radar (SAR) images of real-life military vehicles. SAR data represents imaging in non-visual spectrum and is typically characterized by high noise and low resolution. The proposed supervised ML method is based on sparse representation technique and reduces to convex quadratic and generally p-order cone programming problems. It is shown to deliver excellent results on real-life SAR datasets, as well as on SAR images contaminated with additional noise and occlusions.

Risk-Controlled Wagering Portfolios: Scenario Covering, Downside Protection, and Responsible Sports-Betting DesignThe rapid expansion of legal sports wagering has created a large-scale consumer decision environment in which increasingly complex bets are offered through highly accessible digital platforms. We develop an optimization-based framework for sports wagering that focuses on downside protection rather than expected growth. A bettor allocates a fixed budget across straight bets and multibets while seeking to guarantee a target payout over a prescribed set of outcome scenarios and explicitly controlling exposure to unfavorable outcomes. The resulting model provides a tractable way to study tradeoffs among guaranteed payout, scenario coverage, and risk. We discuss structural properties of optimal wagering strategies, computational approaches, and extensions involving restrictions on parlays and loss probabilities. The framework also provides a basis for studying risk disclosure and responsible-betting mechanisms in increasingly complex wagering environments.

Can an AI Be Your Realtor? Data, Decisions, and Open Problems in Residential Real EstateCan an AI be your realtor? The 2024 NAR settlement unbundled residential commissions and made each component of the agent's role separately priceable. This talk argues that the analytical core of that role — valuation, offer strategy, negotiation preparation — is a sequence of estimation and risk-averse optimization problems under fragmented and adversarially incomplete information. It covers MLS fragmentation and missing-not-at-random inventory as a threat to automated valuation, distributional valuation with tail-constrained bid selection, and the capability and evaluation gaps in current LLM-based agents. Open problems and collaboration opportunities are presented.

Risk-Aware Digital Twins for Biomedical and Engineering Systems: A PDE-Constrained Optimization PerspectiveDigital twins are adaptive virtual representations of physical systems that are updated by data and used to support prediction, monitoring, and decision-making. Their reliability requires more than simulation or machine learning: it requires physics-based models, scalable optimization, uncertainty quantification, and risk-aware decision support. This talk presents a PDE-constrained optimization perspective on trustworthy digital twins, with emphasis on both applications and theoretical foundations. We discuss how digital twins lead to large-scale inverse, control, and design problems constrained by PDEs, and how adjoint methods, inexact trust-region algorithms, and augmented Lagrangian techniques provide scalable tools for model updating and decision-making. A central theme is uncertainty: digital twins must operate with noisy data, incomplete boundary conditions, uncertain parameters, and rare but consequential events. Risk-aware formulations, including CVaR-type objectives and robust perspectives, provide a principled framework for such settings. Applications include patient-specific aneurysm modeling, structural digital twins for bridges, thermal design, sensing, graph-based control, and Maxwell-based systems.

Scalable Generation of Matern Random FieldsRealistic engineering design and optimization require uncertainty models that can represent spatially varying loads, material properties, and geometry without being more computationally expensive than the physics simulation. This presentation demonstrates a scalable framework for generating Matérn Gaussian random fields on large, irregular, meshed domains, where conventional approaches, such as covariance-matrix, Karhunen–Loève, and FFT-based methods, are limited by computational cost, restrictive geometries, or boundary-condition artifacts. The proposed approaches use the stochastic partial differential equation formulation of Matérn fields, which implicitly defines the covariance via a fractional elliptic operator and naturally supports arbitrary domains, boundary conditions, anisotropy, and embedded surfaces or manifolds. Scalability is achieved through two key components: element-local sampling of finite-element white noise with linear computational complexity, and efficient approximation of the fractional operator. Rational approximation techniques, including AAA and BURA, convert the fractional problem to a small number of independent shifted, integer-order finite-element solves that can be executed in parallel using standard multigrid-preconditioned solvers. The resulting algorithm is implemented in MFEM and supports two- and three-dimensional Matérn-type random-field generation on arbitrary meshes. Numerical demonstrations involving heat-sink, bridge, and manifold topology optimization show that the correlation length can significantly alter the optimized structures and that the proposed method makes it computationally feasible to generate hundreds of full-scale stochastic samples per design. The presentation also examines a lower-cost multigrid-based sampler, noting its usefulness when approximate correlation structure is sufficient but its limitations in reproducing an exact target covariance.

Bregman Epi-Regularization of Stochastic Functionals: The Risk Quadrangle FrameworkMany important stochastic optimization and statistical estimation problems involve nonsmooth convex functionals, including conditional value-at-risk, expectile, mean-absolute semideviation, and error measures. Classical epi-regularization (smoothing) approaches typically replace the original nonsmooth objective functional by a fixed (static) smooth approximation. In this talk, we develop an alternative dynamic approach based on Bregman divergences and the Risk Quadrangle (RQ) framework. The central idea is to replace a fixed regularization centered at a nominal reference point by a relative Bregman epi-regularization whose reference point is allowed to evolve during the optimization process. This makes it possible to embed smoothing directly into a Bregman proximal point framework: each proximal step generates a new, locally adapted smooth approximation of the original nonsmooth functional. As a result, the smoothing is no longer external to the optimization algorithm, but becomes part of its dynamics. The proximal interpretation also provides a natural mechanism for updating the reference point and controlling the regularization parameters. Finally, when this construction is embedded into the RQ framework, epi-regularizing a single regret functional automatically induces regularized risk, deviation, error, and statistic functionals. Thus, the approach provides a unified mechanism for adaptive smoothing of stochastic optimization and statistical estimation problems.

Sparse Optimization with Applications to Cancer ResearchThe talk is devoted to investigating single-objective and multiobjective optimization problems involving the ℓ0-norm function, which is nonconvex and nondifferentiable while being variationally convex. Such problems appear in proton beam therapy models of cancer research. The developed approach uses first-order and second-order subdifferential tools of variational analysis and scalarization techniques of multiobjective optimization. Based on this machinery, we propose several algorithms of the subgradient and generalized Newtonian types and conduct their convergence analysis. The obtained results are illustrated by numerical examples from proton therapy models. Based on the collaboration with the Proton Therapy Center of the Corewell William Beaumont Hospital, Royal Oak, Michigan.

Sequential Optimization of Dynamically Augmented CVaRThis talk describes methods for optimization of the Conditional Value-at-Risk (CVaR) of total discounted costs for Markov Decision Processes (MDPs) with finite state and action sets. It introduces the Dynamically augmented CVaR (DCVaR) risk measure and provides an algorithm for its optimization. This paper investigates a specially defined Robust MDP (RMDP), in which the state space is augmented with the tail risk factor. This RMDP, which we call the Dynamically augmented RMDP (DRMDP), was introduced to the literature for calculations of optimal CVaR values by value iteration, but, as was understood later, these value iterations compute lower bounds of minimal static CVaRs. DCVaR is defined as a time consistent version of the static CVaR, and it is a lower bound of the static CVaR. It also can be considered as a dynamic version of the nested CVaR. The correctness of the provided algorithm is proved by studying a special mass transfer problem.

Contextual Risk Measures for Multi-Class Classification: Fairness, Robustness, and Asymptotic AnalysisA new framework for multi-class classification based on coherent systemic risk measures will be presented. The loss functions for each class are evaluated using coherent contextual risk measures, and a systemic risk measure determines the overall classification risk. We construct risk-averse counterparts to a popular multi-class classification method by formulating a two-stage stochastic programming problem for determining the classifier. The proposed approach is also used to address fairness in classification, where the sensitive attributes are modeled as contexts and the overall risk evaluation is obtained by suitable nonlinear class aggregators and a total aggregator. We design a novel risk-averse regularized decomposition method whose computational effort grows linearly with the number of data points. The numerical experiments demonstrate that the proposed framework is particularly effective when the data is noisy, corrupted, or scarce relative to the problem dimension. The risk-averse classifiers exhibit greater robustness and better generalization than their risk-neutral counterparts, with the performance gap widening as the number of classes increases. The method is modified to solve the fairness-aware classifier problem. We demonstrate that the contextual-risk-based classifiers achieve comparable or improved fairness relative to a Wasserstein distributionally robust fair classifier, while maintaining higher predictive performance and lower variability than the existing robust-fair baseline. In a general multi-class and multi-group setting, we also propose a chi-square test, as well as the Gini index for evaluating group fairness. Time-permitting, we shall discuss the statistical consistency of both the total risk estimator and the optimal classifier as the sample size increases. Additionally, we have established a central limit theorem for the systemic risk estimator. This result characterizes the limiting distribution of the estimation error, enabling the construction of confidence intervals and statistical hypothesis tests for the misclassification risk. We illustrate the theoretical findings through numerical experiments on both real image datasets and synthetic data, confirming the predicted convergence to normality. This is joint work with Xiangyu Tian.

Uniform Laws of Large Numbers for SubdifferentialsWe provide counterexamples showing that uniform laws of large numbers do not hold for subdifferentials under natural assumptions. Our results apply to random Lipschitz functions and random convex functions with a finite number of smooth pieces. Consequently, they resolve the questions posed by Shapiro and Xu [J. Math. Anal. Appl., 325(2), 2007] in the negative and highlight the obstacles nonsmoothness poses to uniform results. We also provide positive results giving sufficient conditions for uniform laws to hold. The talk is based on joint work with Dr. Lai Tian.

Risk Quadrangle and Robust Optimization Based on Extended φ-DivergenceThe Fundamental Risk Quadrangle (FRQ) is a unified framework linking risk management, statistical estimation, and optimization. Distributionally robust optimization (DRO) based on φ-divergence minimizes the maximal expected loss, where the maximum is over a φ-divergence ambiguity set. This paper introduces the extended φ-divergence and the extended φ-divergence quadrangle, which integrates DRO into the FRQ framework. We derive the primal and dual representations of the quadrangle elements (risk, deviation, regret, error, and statistic). The dual representation provides an interpretation for classification, portfolio optimization, and regression as robust optimization based on the extended φ-divergence. The primal representation offers tractable formulations of these robust optimizations as convex optimization. We provide illustrative examples showing that many common problems, such as least-squares regression, quantile regression, support vector machines, and CVaR optimization, fall within this framework. We further construct a Hellinger-divergence risk measure and demonstrate its applicability in an empirical portfolio experiment.

Motzkin–Straus Formulation for the Maximum Clique Problem: Minimum Potential Energy Interpretation and ExtensionsThe Motzkin–Straus formulation for the maximum clique problem provides a nontrivial characterization of the clique number of a graph in terms of the maximum value of a nonconvex quadratic function over a standard simplex. It was originally developed as a way of proving Turan's theorem in graph theory but was later used to develop competitive algorithms for the maximum clique problem based on continuous optimization. It is widely regarded as a very important albeit somewhat non-intuitive result. In this presentation, we introduce the concept of “gravitational” potential energy in the context of networks and investigate its implications for interpreting maximum cliques and Motzkin–Straus formulation from a physics perspective. We treat the nodes of a graph as “particles” with masses and consider the Newtonian potential energy of gravitational interactions between them. We show that in these settings, the maximum clique in a graph essentially represents the minimum gravitational potential energy structure. This yields an intuitive physics-based interpretation of the Motzkin–Strauss formulation as the problem of minimizing the Newtonian gravitational potential energy in a graph under certain assumptions. At the end of the presentation, we also mention more recent results on extending the Motzkin–Straus formulation to optimization problems on clique relaxations (namely, maximum s-defective clique and maximum s-plex), which relax the concept of a clique by allowing missing edges.

Cost-Benefit-Aware Network Immunization via Dominant-Eigenvalue OptimizationWe study cost-benefit-aware immunization of large networks based on dominant eigenvalue reduction. The network is represented by a symmetric nonnegative contact-intensity matrix, while each node carries a positive intervention cost. The main model is an epsilon-constrained single-objective formulation: for a given total intervention-cost budget, select nodes for immunization so that the dominant eigenvalue of the retained contact matrix is as small as possible. This formulation generalizes fixed-cardinality node deletion and binary adjacency models. We present an exact mixed-integer semidefinite formulation, a practical mixed-integer linear cutting-plane algorithm based on eigenvector cuts, convex relaxations, optional surrogate integer models, budgeted greedy heuristics, and an exact mixed-integer formulation of the Shield-value proxy. The Shield-value model is included because it gives a quadratic submodular proxy for dominant-eigenvalue reduction and connects the formulation to earlier exact multiobjective immunization work. We also discuss monotonicity, submodularity, and supermodularity, showing by a three-node example that the exact dominant-eigenvalue drop is monotone but neither generally submodular nor generally supermodular. Finally, we report computational experiments using a Gurobi lazy-constraint implementation on Barabasi–Albert and Erdos–Renyi graphs and on two SNAP networks, including brute-force validation on small instances, time-limited synthetic scaling checks up to 1000 nodes, and real-network tests up to 5241 nodes. (The work is coauthored with Dr. Michael Emmerich, the University of Jyvaskyla, Finland)