arXiv subjects

Fatma Kılınç-Karzan

Publications and source records attributed to Fatma Kılınç-Karzan.

17 recordsLinked to original sources

arXiv2026

Parameter-Free Non-Ergodic Extragradient Algorithms for Solving Monotone Variational Inequalities

Monotone variational inequalities (VIs) provide a unifying framework for convex minimization, equilibrium computation, and convex-concave saddle-point problems. Extragradient-type methods are among the most effective first-order algorithms for such problems, but their performance hinges critically on stepsize selection. While most existing theory focuses on ergodic averages of the iterates, practical performance is often driven by the significantly stronger behavior of the last iterate. Moreover, available last-iterate guarantees typically rely on fixed stepsizes chosen using problem-specific global smoothness information, which is often difficult to estimate accurately and may not even be applicable. In this paper, we develop parameter-free extragradient methods with non-asymptotic last-iterate guarantees for constrained monotone VIs. For globally Lipschitz operators, our algorithm achieves an $o(1/\sqrt{T})$ last-iterate rate. We then extend the framework to locally Lipschitz operators via backtracking line search and obtain the same rate while preserving parameter-freeness, thereby making parameter-free last-iterate methods applicable to important problem classes for which global smoothness is unrealistic. Our numerical experiments on bilinear matrix games, LASSO, minimax group fairness, and state-of-the-art maximum entropy sampling relaxations demonstrate wide applicability of our results as well as strong last-iterate performance and significant improvements over existing methods.

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arXiv2026

From Majorization to Scaling: Advancing Convex Relaxations of Maximum Entropy Sampling Problem

In this paper, we study the maximum entropy sampling problem (MESP) and its variants. MESP seeks to identify a small subset of variables that maximizes the determinant of a covariance submatrix, and is a fundamental model in optimal experimental design and information acquisition. Although MESP is combinatorial and NP-hard, continuous relaxations, most notably linx and $Γ$ factorization, provide tractable approximations, yet their derivation, relative strength, and potential for systematic improvement remain poorly understood. We address this gap by introducing two main ideas: a unified majorization-based framework for deriving and analyzing relaxations, and a novel scaling-based bound-enhancement technique, which we call double-scaling. Our approach is motivated by the observation that the difficulty of MESP arises from two distinct sources: the combinatorial selection structure and the lack of permutation symmetry in the spectral objective. Majorization naturally resolves the latter by symmetrizing the spectral function and yielding its convex envelope. In the log-determinant setting, we establish the main theoretical properties of double-scaling and prove that it strictly dominates previously known scaling bounds. Using our majorization-based alternative characterization of $Γ$ factorization relaxation, we also derive, for the first time, formal dominance relations between linx- and $Γ$ factorization-bounds, as well as between their scaling-strengthened variants. Our numerical results show that our double-scaled linx relaxation consistently and substantially outperforms existing scaling methods and compares quite favorably with other state-of-the-art relaxations in terms of both bound quality and computational efficiency.

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arXiv2025

Convergence, Duality and Well-Posedness in Convex Bilevel Optimization

We consider the convex bilevel optimization problem, also known as simple bilevel programming. There are two challenges in solving convex bilevel optimization problems. Firstly, strong duality is not guaranteed due to the lack of Slater constraint qualification. Secondly, we demonstrate through an example that convergence of algorithms is not guaranteed even when usual subotimality gap bounds are present, due to the possibility of encountering super-optimal solutions. We show that strong duality (but not necessarily dual solvability) is exactly equivalent to ensuring correct asymptotic convergence of both inner and outer function values, and provide a simple condition that guarantees strong duality. Unfortunately, we also show that this simple condition is not sufficient to guarantee convergence to the optimal solution set. We draw connections to Levitin-Polyak well-posedness, and leverage this together with our strong duality equivalence to provide another condition that ensures convergence to the optimal solution set. We also discuss how our conditions have been implicitly present in existing algorithmic work.

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arXiv2025

Efficient Online Large-Margin Classification via Dual Certificates

Online classification is a central problem in optimization, statistical learning and data science. Classical algorithms such as the perceptron offer efficient updates and finite mistake guarantees on linearly separable data, but they do not exploit the underlying geometric structure of the classification problem. We study the offline maximum margin problem through its dual formulation and use the resulting geometric insights to design a principled and efficient algorithm for the online setting. A key feature of our method is its translation invariance, inherited from the offline formulation, which plays a central role in its performance analysis. Our theoretical analysis yields improved mistake and margin bounds that depend only on translation-invariant quantities, offering stronger guarantees than existing algorithms under the same assumptions in favorable settings. In particular, we identify a parameter regime where our algorithm makes at most two mistakes per sequence, whereas the perceptron can be forced to make arbitrarily many mistakes. Our numerical study on real data further demonstrates that our method matches the computational efficiency of existing online algorithms, while significantly outperforming them in accuracy.

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arXiv2024

On the strength of Burer's lifted convex relaxation to quadratic programming with ball constraints

We study quadratic programs with $m$ ball constraints, and the strength of a lifted convex relaxation for it recently proposed by Burer (2024). Burer shows this relaxation is exact when $m=2$. For general $m$, Burer (2024) provides numerical evidence that this lifted relaxation is tighter than the Kronecker product based Reformulation Linearization Technique (RLT) inequalities introduced by Anstreicher (2017), and conjectures that this must be theoretically true as well. In this note, we provide an affirmative answer to this question and formally prove that this lifted relaxation indeed implies the Kronecker inequalities. Our proof is based on a decomposition of non-rank-one extreme rays of the lifted relaxation for each pair of ball constraints. Burer (2024) also numerically observes that for this lifted relaxation, an RLT-based inequality proposed by Zhen et al. (2021) is redundant, and conjectures this to be theoretically true as well. We also provide a formal proof that Zhen et al. (2021) inequalities are redundant for this lifted relaxation. In addition, we establish that Burer's lifted relaxation is a particular case of the moment-sum-of-squares hierarchy.

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arXiv2024

Mistake, Manipulation and Margin Guarantees in Online Strategic Classification

We consider an online strategic classification problem where each arriving agent can manipulate their true feature vector to obtain a positive predicted label, while incurring a cost that depends on the amount of manipulation. The learner seeks to predict the agent's true label given access to only the manipulated features. After the learner releases their prediction, the agent's true label is revealed. Previous algorithms such as the strategic perceptron guarantee finitely many mistakes under a margin assumption on agents' true feature vectors. However, these are not guaranteed to encourage agents to be truthful. Promoting truthfulness is intimately linked to obtaining adequate margin on the predictions, thus we provide two new algorithms aimed at recovering the maximum margin classifier in the presence of strategic agent behavior. We prove convergence, finite mistake and finite manipulation guarantees for a variety of agent cost structures. We also provide generalized versions of the strategic perceptron with mistake guarantees for different costs. Our numerical study on real and synthetic data demonstrates that the new algorithms outperform previous ones in terms of margin, number of manipulation and number of mistakes.

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arXiv2023

Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications

We consider a general conic mixed-binary set where each homogeneous conic constraint $j$ involves an affine function of independent continuous variables and an epigraph variable associated with a nonnegative function, $f_j$, of common binary variables. Sets of this form naturally arise as substructures in a number of applications including mean-risk optimization, chance-constrained problems, portfolio optimization, lot-sizing and scheduling, fractional programming, variants of the best subset selection problem, a class of sparse semidefinite programs, and distributionally robust chance-constrained programs. We give a convex hull description of this set that relies on simultaneous characterization of the epigraphs of $f_j$'s, which is easy to do when all functions $f_j$'s are submodular. Our result unifies and generalizes an existing result in two important directions. First, it considers \emph{multiple general convex cone} constraints instead of a single second-order cone type constraint. Second, it takes \emph{arbitrary nonnegative functions} instead of a specific submodular function obtained from the square root of an affine function. We close by demonstrating the applicability of our results in the context of a number of problem classes.

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arXiv2023

Constrained Optimization of Rank-One Functions with Indicator Variables

Optimization problems involving minimization of a rank-one convex function over constraints modeling restrictions on the support of the decision variables emerge in various machine learning applications. These problems are often modeled with indicator variables for identifying the support of the continuous variables. In this paper we investigate compact extended formulations for such problems through perspective reformulation techniques. In contrast to the majority of previous work that relies on support function arguments and disjunctive programming techniques to provide convex hull results, we propose a constructive approach that exploits a hidden conic structure induced by perspective functions. To this end, we first establish a convex hull result for a general conic mixed-binary set in which each conic constraint involves a linear function of independent continuous variables and a set of binary variables. We then demonstrate that extended representations of sets associated with epigraphs of rank-one convex functions over constraints modeling indicator relations naturally admit such a conic representation. This enables us to systematically give perspective formulations for the convex hull descriptions of these sets with nonlinear separable or non-separable objective functions, sign constraints on continuous variables, and combinatorial constraints on indicator variables. We illustrate the efficacy of our results on sparse nonnegative logistic regression problems.

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arXiv2023

A Field Test of Bandit Algorithms for Recommendations: Understanding the Validity of Assumptions on Human Preferences in Multi-armed Bandits

Personalized recommender systems suffuse modern life, shaping what media we read and what products we consume. Algorithms powering such systems tend to consist of supervised learning-based heuristics, such as latent factor models with a variety of heuristically chosen prediction targets. Meanwhile, theoretical treatments of recommendation frequently address the decision-theoretic nature of the problem, including the need to balance exploration and exploitation, via the multi-armed bandits (MABs) framework. However, MAB-based approaches rely heavily on assumptions about human preferences. These preference assumptions are seldom tested using human subject studies, partly due to the lack of publicly available toolkits to conduct such studies. In this work, we conduct a study with crowdworkers in a comics recommendation MABs setting. Each arm represents a comic category, and users provide feedback after each recommendation. We check the validity of core MABs assumptions-that human preferences (reward distributions) are fixed over time-and find that they do not hold. This finding suggests that any MAB algorithm used for recommender systems should account for human preference dynamics. While answering these questions, we provide a flexible experimental framework for understanding human preference dynamics and testing MABs algorithms with human users. The code for our experimental framework and the collected data can be found at https://github.com/HumainLab/human-bandit-evaluation.

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arXiv2021

Exactness in SDP relaxations of QCQPs: Theory and applications

Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems. In a QCQP, we are asked to minimize a (possibly nonconvex) quadratic function subject to a number of (possibly nonconvex) quadratic constraints. Such problems arise naturally in many areas of operations research, computer science, and engineering. Although QCQPs are NP-hard to solve in general, they admit a natural convex relaxation via the standard (Shor) semidefinite program (SDP) relaxation. In this tutorial, we will study the SDP relaxation for general QCQPs, present various exactness concepts related to this relaxation and discuss conditions guaranteeing such SDP exactness. In particular, we will define and examine three notions of SDP exactness: (i) objective value exactness -- the condition that the optimal value of the QCQP and the optimal value of its SDP relaxation coincide, (ii) convex hull exactness -- the condition that the convex hull of the QCQP epigraph coincides with the (projected) SDP epigraph, and (iii) the rank-one generated (ROG) property -- the condition that a particular conic subset of the positive semidefinite matrices related to a given QCQP is generated by its rank-one matrices. Our analysis for objective value exactness and convex hull exactness stems from a geometric treatment of the projected SDP relaxation and crucially considers how the objective function interacts with the constraints. The ROG property complements these results by offering a sufficient condition for both objective value exactness and convex hull exactness which is oblivious to the objective function. We will give a variety of sufficient conditions for these exactness conditions and discuss settings where these sufficient conditions are additionally necessary. Throughout, we will highlight implications of our results for a number of example applications.

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arXiv2021

Joint chance-constrained programs and the intersection of mixing sets through a submodularity lens

A particularly important substructure in modeling joint linear chance-constrained programs with random right-hand sides and finite sample space is the intersection of mixing sets with common binary variables (and possibly a knapsack constraint). In this paper, we first revisit basic mixing sets by establishing a strong and previously unrecognized connection to submodularity. In particular, we show that mixing inequalities with binary variables are nothing but the polymatroid inequalities associated with a specific submodular function. This submodularity viewpoint enables us to unify and extend existing results on valid inequalities and convex hulls of the intersection of multiple mixing sets with common binary variables. Then, we study such intersections under an additional linking constraint lower bounding a linear function of the continuous variables. This is motivated from the desire to exploit the information encoded in the knapsack constraint arising in joint linear CCPs via the quantile cuts. We propose a new class of valid inequalities and characterize when this new class along with the mixing inequalities are sufficient to describe the convex hull.

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arXiv2021

Online Convex Optimization Perspective for Learning from Dynamically Revealed Preferences

We study the problem of online learning (OL) from revealed preferences: a learner wishes to learn a non-strategic agent's private utility function through observing the agent's utility-maximizing actions in a changing environment. We adopt an online inverse optimization setup, where the learner observes a stream of agent's actions in an online fashion and the learning performance is measured by regret associated with a loss function. We first characterize a special but broad class of agent's utility functions, then utilize this structure in designing a new convex loss function. We establish that the regret with respect to our new loss function also bounds the regret with respect to all other usual loss functions in the literature. This allows us to design a flexible OL framework that enables a unified treatment of loss functions and supports a variety of online convex optimization algorithms. We demonstrate with theoretical and empirical evidence that our framework based on the new loss function (in particular online Mirror Descent) has significant advantages in terms of regret performance and solution time over other OL algorithms from the literature and bypasses the previous technical assumptions as well.

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arXiv2021

Necessary and sufficient conditions for rank-one generated cones

A closed convex conic subset $\mathcal{S}$ of the positive semidefinite (PSD) cone is rank-one generated (ROG) if all of its extreme rays are generated by rank-one matrices. The ROG property of $\mathcal{S}$ is closely related to the exactness of SDP relaxations of nonconvex quadratically constrained quadratic programs (QCQPs) related to $\mathcal{S}$. We consider the case where $\mathcal{S}$ is obtained as the intersection of the PSD cone with finitely many homogeneous linear matrix inequalities and conic constraints and identify sufficient conditions that guarantee that $\mathcal{S}$ is ROG. Our general framework allows us to recover a number of well-known results from the literature. In the case of two linear matrix inequalities, we also establish the necessity of our sufficient conditions. This extends one of the few settings from the literature -- the case of one linear matrix inequality and the S-lemma -- where an explicit characterization for the ROG property exists. Finally, we show how our ROG results on cones can be translated into inhomogeneous SDP exactness results and convex hull descriptions in the original space of a QCQP. We close with a few applications of these results; specifically, we recover the well-known perspective reformulation of a simple mixed-binary set via the ROG toolkit.

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arXiv2021

Strong Formulations for Distributionally Robust Chance-Constrained Programs with Left-Hand Side Uncertainty under Wasserstein Ambiguity

Distributionally robust chance-constrained programs (DR-CCP) over Wasserstein ambiguity sets exhibit attractive out-of-sample performance and admit big-$M$-based mixed-integer programming (MIP) reformulations with conic constraints. However, the resulting formulations often suffer from scalability issues as sample size increases. To address this shortcoming, we derive stronger formulations that scale well with respect to the sample size. Our focus is on ambiguity sets under the so-called left-hand side (LHS) uncertainty, where the uncertain parameters affect the coefficients of the decision variables in the linear inequalities defining the safety sets. The interaction between the uncertain parameters and the variable coefficients in the safety set definition causes challenges in strengthening the original big-$M$ formulations. By exploiting the connection between nominal chance-constrained programs and DR-CCP, we obtain strong formulations with significant enhancements. In particular, through this connection, we derive a linear number of valid inequalities, which can be immediately added to the formulations to obtain improved formulations in the original space of variables. In addition, we suggest a quantile-based strengthening procedure that allows us to reduce the big-$M$ coefficients drastically. Furthermore, based on this procedure, we propose an exponential class of inequalities that can be separated efficiently within a branch-and-cut framework. The quantile-based strengthening procedure can be expensive. Therefore, for the special case of covering and packing type problems, we identify an efficient scheme to carry out this procedure. We demonstrate the computational efficacy of our proposed formulations on two classes of problems, namely stochastic portfolio optimization and resource planning.

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arXiv2020

Risk Guarantees for End-to-End Prediction and Optimization Processes

Prediction models are often employed in estimating parameters of optimization models. Despite the fact that in an end-to-end view, the real goal is to achieve good optimization performance, the prediction performance is measured on its own. While it is usually believed that good prediction performance in estimating the parameters will result in good subsequent optimization performance, formal theoretical guarantees on this are notably lacking. In this paper, we explore conditions that allow us to explicitly describe how the prediction performance governs the optimization performance. Our weaker condition allows for an asymptotic convergence result, while our stronger condition allows for exact quantification of the optimization performance in terms of the prediction performance. In general, verification of these conditions is a non-trivial task. Nevertheless, we show that our weaker condition is equivalent to the well-known Fisher consistency concept from the learning theory literature. This then allows us to easily check our weaker condition for several loss functions. We also establish that the squared error loss function satisfies our stronger condition. Consequently, we derive the exact theoretical relationship between prediction performance measured with the squared loss, as well as a class of symmetric loss functions, and the subsequent optimization performance. In a computational study on portfolio optimization, fractional knapsack and multiclass classification problems, we compare the optimization performance of using of several prediction loss functions (some that are Fisher consistent and some that are not) and demonstrate that lack of consistency of the loss function can indeed have a detrimental effect on performance.

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arXiv2020

Distributionally Robust Chance-Constrained Programs with Right-Hand Side Uncertainty under Wasserstein Ambiguity

We consider exact deterministic mixed-integer programming (MIP) reformulations of distributionally robust chance-constrained programs (DR-CCP) with random right-hand sides over Wasserstein ambiguity sets. The existing MIP formulations are known to have weak continuous relaxation bounds, and, consequently, for hard instances with small radius, or with large problem sizes, the branch-and-bound based solution processes suffer from large optimality gaps even after hours of computation time. This significantly hinders the practical application of the DR-CCP paradigm. Motivated by these challenges, we conduct a polyhedral study to strengthen these formulations. We reveal several hidden connections between DR-CCP and its nominal counterpart (the sample average approximation), mixing sets, and robust 0-1 programming. By exploiting these connections in combination, we provide an improved formulation and two classes of valid inequalities for DR-CCP. We test the impact of our results on a stochastic transportation problem numerically. Our experiments demonstrate the effectiveness of our approach; in particular our improved formulation and proposed valid inequalities reduce the overall solution times remarkably. Moreover, this allows us to significantly scale up the problem sizes that can be handled in such DR-CCP formulations by reducing the solution times from hours to seconds.

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arXiv2015

On Minimal Valid Inequalities for Mixed Integer Conic Programs

We study disjunctive conic sets involving a general regular (closed, convex, full dimensional, and pointed) cone K such as the nonnegative orthant, the Lorentz cone or the positive semidefinite cone. In a unified framework, we introduce K-minimal inequalities and show that under mild assumptions, these inequalities together with the trivial cone-implied inequalities are sufficient to describe the convex hull. We study the properties of K-minimal inequalities by establishing algebraic necessary conditions for an inequality to be K-minimal. This characterization leads to a broader algebraically defined class of K- sublinear inequalities. We establish a close connection between K-sublinear inequalities and the support functions of sets with a particular structure. This connection results in practical ways of showing that a given inequality is K-sublinear and K-minimal. Our framework generalizes some of the results from the mixed integer linear case. It is well known that the minimal inequalities for mixed integer linear programs are generated by sublinear (positively homogeneous, subadditive and convex) functions that are also piecewise linear. This result is easily recovered by our analysis. Whenever possible we highlight the connections to the existing literature. However, our study unveils that such a cut generating function view treating the data associated with each individual variable independently is not possible in the case of general cones other than nonnegative orthant, even when the cone involved is the Lorentz cone.

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