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Fatma Kilinc-Karzan

Publications and source records attributed to Fatma Kilinc-Karzan.

15 recordsLinked to original sources
arXiv2024

On semidefinite descriptions for convex hulls of quadratic programs

Quadratically constrained quadratic programs (QCQPs) are a highly expressive class of nonconvex optimization problems. While QCQPs are NP-hard in general, they admit a natural convex relaxation via the standard semidefinite program (SDP) relaxation. In this paper we study when the convex hull of the epigraph of a QCQP coincides with the projected epigraph of the SDP relaxation. We present a sufficient condition for convex hull exactness and show that this condition is further necessary under an additional geometric assumption. The sufficient condition is based on geometric properties of $Γ$, the cone of convex Lagrange multipliers, and its relatives $Γ_1$ and $Γ^\circ$.

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arXiv2024

Accelerated first-order methods for a class of semidefinite programs

This paper introduces a new storage-optimal first-order method (FOM), CertSDP, for solving a special class of semidefinite programs (SDPs) to high accuracy. The class of SDPs that we consider, the exact QMP-like SDPs, is characterized by low-rank solutions, a priori knowledge of the restriction of the SDP solution to a small subspace, and standard regularity assumptions such as strict complementarity. Crucially, we show how to use a certificate of strict complementarity to construct a low-dimensional strongly convex minimax problem whose optimizer coincides with a factorization of the SDP optimizer. From an algorithmic standpoint, we show how to construct the necessary certificate and how to solve the minimax problem efficiently. We accompany our theoretical results with preliminary numerical experiments suggesting that CertSDP significantly outperforms current state-of-the-art methods on large sparse exact QMP-like SDPs.

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arXiv2021

Implicit regularity and linear convergence rates for the generalized trust-region subproblem

In this paper we develop efficient first-order algorithms for the generalized trust-region subproblem (GTRS), which has applications in signal processing, compressed sensing, and engineering. Although the GTRS, as stated, is nonlinear and nonconvex, it is well-known that objective value exactness holds for its SDP relaxation under a Slater condition. While polynomial-time SDP-based algorithms exist for the GTRS, their relatively large computational complexity has motivated and spurred the development of custom approaches for solving the GTRS. In particular, recent work in this direction has developed first-order methods for the GTRS whose running times are linear in the sparsity (the number of nonzero entries) of the input data. In contrast to these algorithms, in this paper we develop algorithms for computing $ε$-approximate solutions to the GTRS whose running times are linear in both the input sparsity and the precision $\log(1/ε)$ whenever a regularity parameter is positive. We complement our theoretical guarantees with numerical experiments comparing our approach against algorithms from the literature. Our numerical experiments highlight that our new algorithms significantly outperform prior state-of-the-art algorithms on sparse large-scale instances.

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arXiv2021

A Geometric View of SDP Exactness in QCQPs and its Applications

Quadratically constrained quadratic programs (QCQPs) are a highly expressive class of nonconvex optimization problems. While QCQPs are NP-hard in general, they admit a natural convex relaxation via the standard (Shor) semidefinite program (SDP) relaxation. Towards understanding when this relaxation is exact, we study general QCQPs and their (projected) SDP relaxations. We present sufficient (and in some cases, also necessary) conditions for objective value exactness (the condition that the objective values of the QCQP and its SDP relaxation coincide) and convex hull exactness (the condition that the convex hull of the QCQP epigraph coincides with the epigraph of its SDP relaxation). Our conditions for exactness are based on geometric properties of $Γ$, the cone of convex Lagrange multipliers, and its relatives $Γ_P$ and $Γ^\circ$. These tools form the basis of our main message: questions of exactness can be treated systematically whenever $Γ$, $Γ_P$, or $Γ^\circ$ is well-understood. As further evidence of this message, we apply our tools to address questions of exactness for a prototypical QCQP involving a binary on-off constraint, quadratic matrix programs, the QCQP formulation of the partition problem, and random and semi-random QCQPs.

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arXiv2021

Rebounding Bandits for Modeling Satiation Effects

Psychological research shows that enjoyment of many goods is subject to satiation, with short-term satisfaction declining after repeated exposures to the same item. Nevertheless, proposed algorithms for powering recommender systems seldom model these dynamics, instead proceeding as though user preferences were fixed in time. In this work, we introduce rebounding bandits, a multi-armed bandit setup, where satiation dynamics are modeled as time-invariant linear dynamical systems. Expected rewards for each arm decline monotonically with consecutive exposures to it and rebound towards the initial reward whenever that arm is not pulled. Unlike classical bandit settings, methods for tackling rebounding bandits must plan ahead and model-based methods rely on estimating the parameters of the satiation dynamics. We characterize the planning problem, showing that the greedy policy is optimal when the arms exhibit identical deterministic dynamics. To address stochastic satiation dynamics with unknown parameters, we propose Explore-Estimate-Plan (EEP), an algorithm that pulls arms methodically, estimates the system dynamics, and then plans accordingly.

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arXiv2020

On the tightness of SDP relaxations of QCQPs

Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study conditions under which the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the SDP relaxation is tight whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. We present similar sufficient conditions under which the projected epigraph of the SDP gives the convex hull of the epigraph in the original QCQP. Our results also imply new sufficient conditions for the tightness (as well as convex hull exactness) of a second order cone program relaxation of simultaneously diagonalizable QCQPs.

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arXiv2020

The Generalized Trust Region Subproblem: solution complexity and convex hull results

We consider the Generalized Trust Region Subproblem (GTRS) of minimizing a nonconvex quadratic objective over a nonconvex quadratic constraint. A lifting of this problem recasts the GTRS as minimizing a linear objective subject to two nonconvex quadratic constraints. Our first main contribution is structural: we give an explicit description of the convex hull of this nonconvex set in terms of the generalized eigenvalues of an associated matrix pencil. This result may be of interest in building relaxations for nonconvex quadratic programs. Moreover, this result allows us to reformulate the GTRS as the minimization of two convex quadratic functions in the original space. Our next set of contributions is algorithmic: we present an algorithm for solving the GTRS up to an epsilon additive error based on this reformulation. We carefully handle numerical issues that arise from inexact generalized eigenvalue and eigenvector computations and establish explicit running time guarantees for these algorithms. Notably, our algorithms run in linear (in the size of the input) time. Furthermore, our algorithm for computing an epsilon-optimal solution has a slightly-improved running time dependence on epsilon over the state-of-the-art algorithm. Our analysis shows that the dominant cost in solving the GTRS lies in solving a generalized eigenvalue problem -- establishing a natural connection between these problems. Finally, generalizations of our convex hull results allow us to apply our algorithms and their theoretical guarantees directly to equality-, interval-, and hollow- constrained variants of the GTRS. This gives the first linear-time algorithm in the literature for these variants of the GTRS.

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arXiv2020

Dynamic Data-Driven Estimation of Non-Parametric Choice Models

We study non-parametric estimation of choice models, which were introduced to alleviate unreasonable assumptions in traditional parametric models, and are prevalent in several application areas. Existing literature focuses only on the static observational setting where all of the observations are given upfront, they are not equipped with explicit convergence rate guarantees, and consequently they cannot provide an a priori analysis for the model accuracy vs sparsity trade-off on the actual estimated model returned by their algorithms. As opposed to this, we focus on estimating a non-parametric choice model from observational data in a \emph{dynamic} setting, where observations are obtained over time. We show that choice model estimation can be cast as a convex-concave saddle-point (SP) joint estimation and optimization (JEO) problem, and we provide a primal-dual framework for deriving algorithms to solve this based on online convex optimization. By tailoring our framework carefully to the choice model estimation problem, we obtain tractable algorithms with provable convergence guarantees and explicit bounds on the sparsity of the estimated model. Our numerical experiments confirm the effectiveness of the algorithms derived from our framework.

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arXiv2020

On convex hulls of epigraphs of QCQPs

Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study sufficient conditions for a convex hull result that immediately implies that the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the convex hull result holds whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. Our results also imply new sufficient conditions for the tightness (as well as convex hull exactness) of a second order cone program relaxation of simultaneously diagonalizable QCQPs.

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arXiv2018

Exploiting Problem Structure in Optimization under Uncertainty via Online Convex Optimization

In this paper, we consider two paradigms that are developed to account for uncertainty in optimization models: robust optimization (RO) and joint estimation-optimization (JEO). We examine recent developments on efficient and scalable iterative first-order methods for these problems, and show that these iterative methods can be viewed through the lens of online convex optimization (OCO). The standard OCO framework has seen much success for its ability to handle decision-making in dynamic, uncertain, and even adversarial environments. Nevertheless, our applications of interest present further flexibility in OCO via three simple modifications to standard OCO assumptions: we introduce two new concepts of weighted regret and online saddle point problems and study the possibility of making lookahead (anticipatory) decisions. Our analyses demonstrate that these flexibilities introduced into the OCO framework have significant consequences whenever they are applicable. For example, in the strongly convex case, minimizing unweighted regret has a proven optimal bound of $O(\log(T)/T)$, whereas we show that a bound of $O(1/T)$ is possible when we consider weighted regret. Similarly, for the smooth case, considering $1$-lookahead decisions results in a $O(1/T)$ bound, compared to $O(1/\sqrt{T})$ in the standard OCO setting. Consequently, these OCO tools are instrumental in exploiting structural properties of functions and resulting in improved convergence rates for RO and JEO. In certain cases, our results for RO and JEO match the best known or optimal rates in the corresponding problem classes without data uncertainty.

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arXiv2017

Online First-Order Framework for Robust Convex Optimization

Robust optimization (RO) has emerged as one of the leading paradigms to efficiently model parameter uncertainty. The recent connections between RO and problems in statistics and machine learning domains demand for solving RO problems in ever more larger scale. However, the traditional approaches for solving RO formulations based on building and solving robust counterparts or the iterative approaches utilizing nominal feasibility oracles can be prohibitively expensive and thus significantly hinder the scalability of RO paradigm. In this paper, we present a general and flexible iterative framework to approximately solve robust convex optimization problems that is built on a fully online first-order paradigm. In comparison to the existing literature, a key distinguishing feature of our approach is that it only requires access to first-order oracles that are remarkably cheaper than pessimization or nominal feasibility oracles, while maintaining the same convergence rates. This, in particular, makes our approach much more scalable and hence preferable in large-scale applications, specifically those from machine learning and statistics domains. We also provide new interpretations of existing iterative approaches in our framework and illustrate our framework on robust quadratic programming.

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arXiv2017

Theoretical and Practical Advances on Smoothing for Extensive-Form Games

Sparse iterative methods, in particular first-order methods, are known to be among the most effective in solving large-scale two-player zero-sum extensive-form games. The convergence rates of these methods depend heavily on the properties of the distance-generating function that they are based on. We investigate the acceleration of first-order methods for solving extensive-form games through better design of the dilated entropy function---a class of distance-generating functions related to the domains associated with the extensive-form games. By introducing a new weighting scheme for the dilated entropy function, we develop the first distance-generating function for the strategy spaces of sequential games that has no dependence on the branching factor of the player. This result improves the convergence rate of several first-order methods by a factor of $Ω(b^dd)$, where $b$ is the branching factor of the player, and $d$ is the depth of the game tree. Thus far, counterfactual regret minimization methods have been faster in practice, and more popular, than first-order methods despite their theoretically inferior convergence rates. Using our new weighting scheme and practical tuning we show that, for the first time, the excessive gap technique can be made faster than the fastest counterfactual regret minimization algorithm, CFR+, in practice.

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arXiv2016

A Second-Order Cone Based Approach for Solving the Trust Region Subproblem and Its Variants

We study the trust-region subproblem (TRS) of minimizing a nonconvex quadratic function over the unit ball with additional conic constraints. Despite having a nonconvex objective, it is known that the classical TRS and a number of its variants are polynomial-time solvable. In this paper, we follow a second-order cone (SOC) based approach to derive an exact convex reformulation of the TRS under a structural condition on the conic constraint. Our structural condition is immediately satisfied when there is no additional conic constraints, and it generalizes several such conditions studied in the literature. As a result, our study highlights an explicit connection between the classical nonconvex TRS and smooth convex quadratic minimization, which allows for the application of cheap iterative methods such as Nesterov's accelerated gradient descent, to the TRS. Furthermore, under slightly stronger conditions, we give a low-complexity characterization of the convex hull of the epigraph of the nonconvex quadratic function intersected with the constraints defining the domain without any additional variables. We also explore the inclusion of additional hollow constraints to the domain of the TRS, and convexification of the associated epigraph.

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arXiv2016

How to Convexify the Intersection of a Second Order Cone and a Nonconvex Quadratic

A recent series of papers has examined the extension of disjunctive-programming techniques to mixed-integer second-order-cone programming. For example, it has been shown---by several authors using different techniques---that the convex hull of the intersection of an ellipsoid, $E$, and a split disjunction, $(l - x_j)(x_j - u) \le 0$ with $l < u$, equals the intersection of $E$ with an additional second-order-cone representable (SOCr) set. In this paper, we study more general intersections of the form $K \cap Q$ and $K \cap Q \cap H$, where $K$ is a SOCr cone, $Q$ is a nonconvex cone defined by a single homogeneous quadratic, and $H$ is an affine hyperplane. Under several easy-to-verify conditions, we derive simple, computable convex relaxations $K \cap S$ and $K \cap S \cap H$, where $S$ is a SOCr cone. Under further conditions, we prove that these two sets capture precisely the corresponding conic/convex hulls. Our approach unifies and extends previous results, and we illustrate its applicability and generality with many examples.

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arXiv2014

Two-Term Disjunctions on the Second-Order Cone

Balas introduced disjunctive cuts in the 1970s for mixed-integer linear programs. Several recent papers have attempted to extend this work to mixed-integer conic programs. In this paper we study the structure of the convex hull of a two-term disjunction applied to the second-order cone, and develop a methodology to derive closed-form expressions for convex inequalities describing the resulting convex hull. Our approach is based on first characterizing the structure of undominated valid linear inequalities for the disjunction and then using conic duality to derive a family of convex, possibly nonlinear, valid inequalities that correspond to these linear inequalities. We identify and study the cases where these valid inequalities can equivalently be expressed in conic quadratic form and where a single inequality from this family is sufficient to describe the convex hull. In particular, our results on two-term disjunctions on the second-order cone generalize related results on split cuts by Modaresi, Kilinc, and Vielma, and by Andersen and Jensen.

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