About

Fatma Kılınç-Karzan is a Professor of Operations Research at the Tepper School of Business, Carnegie Mellon University. Her role involves teaching and research within the faculty, contributing to the academic community at CMU. The Tepper School emphasizes experiential learning and practical application across its programs, and Professor Kılınç-Karzan is part of this environment, engaging in research and education that align with the school's strategic focus on the intersection of business, technology, and analytics.

Research topics

• Computer Science
• Algorithm
• Mathematics
• Mathematical optimization

Selected publications

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

  ArXiv.org · 2026-04-09

  articleOpen accessSenior author

  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.

  PublisherOA PDF

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

  arXiv (Cornell University) · 2026-04-11

  articleOpen accessSenior author

  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.

  PublisherOA PDF

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

  arXiv (Cornell University) · 2026-04-11

  preprintOpen accessSenior author

  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.

  PublisherDOI

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

  arXiv (Cornell University) · 2026-04-09

  preprintOpen accessSenior author

  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.

  PublisherDOI

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

  Mathematical Programming · 2025-09-09 · 1 citations

  articleOpen access1st author

  Abstract 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 $$m=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . 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 in the original space. 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)’s as well as Jiang and Li (2019)’s SST 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.

  PublisherOA PDFDOI

• Essential Mathematics for Convex Optimization

  Cambridge University Press eBooks · 2025-06-26

  book1st authorCorresponding

  With an emphasis on timeless essential mathematical background for optimization, this textbook provides a comprehensive and accessible introduction to convex optimization for students in applied mathematics, computer science, and engineering. Authored by two influential researchers, the book covers both convex analysis basics and modern topics such as conic programming, conic representations of convex sets, and cone-constrained convex problems, providing readers with a solid, up-to-date understanding of the field. By excluding modeling and algorithms, the authors are able to discuss the theoretical aspects in greater depth. Over 170 in-depth exercises provide hands-on experience with the theory, while more than 30 'Facts' and their accompanying proofs enhance approachability. Instructors will appreciate the appendices that cover all necessary background and the instructors-only solutions manual provided online. By the end of the book, readers will be well equipped to engage with state-of-the-art developments in optimization and its applications in decision-making and engineering.

  PublisherDOI

• Convergence, Duality and Well-Posedness in Convex Bilevel Optimization

  ArXiv.org · 2025-09-22

  preprintOpen access

  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.

  PublisherOA PDFDOI

• On semidefinite descriptions for convex hulls of quadratic programs

  Operations Research Letters · 2024-03-07 · 1 citations

  articleSenior author
  PublisherDOI

• Accelerated first-order methods for a class of semidefinite programs

  Mathematical Programming · 2024-03-22 · 1 citations

  articleSenior author
  PublisherDOI

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

  arXiv (Cornell University) · 2024-07-20

  preprintOpen access1st authorCorresponding

  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.

  PublisherOA PDFDOI

Frequent coauthors

• Alex L. Wang

  Carnegie Mellon University

  22 shared

• Nam Ho-Nguyen

  University of Sydney

  16 shared

• Si̇mge Küçükyavuz

  Northwestern University

  10 shared

• Dabeen Lee

  9 shared

• Tüomas Sandholm

  Carnegie Mellon University

  5 shared

• Kevin Waugh

  The Open University

  5 shared

• Christian Kroer

  5 shared

• Yunlei Lu

  Peking University

  4 shared