Alper Atamtürk

Alicia Y. Tsai, Juliette Decugis, Laurent El Ghaoui et al.

Implicit models are a general class of learning models that forgo the hierarchical layer structure typical in neural networks and instead define the internal states based on an ``equilibrium'' equation, offering competitive performance and reduced memory consumption. However, training such models usually relies on expensive implicit differentiation for backward propagation. In this work, we present a new approach to training implicit models, called State-driven Implicit Modeling (SIM), where we constrain the internal states and outputs to match that of a baseline model, circumventing costly backward computations. The training problem becomes convex by construction and can be solved in a parallel fashion, thanks to its decomposable structure. We demonstrate how the SIM approach can be applied to significantly improve sparsity (parameter reduction) and robustness of baseline models trained on FashionMNIST and CIFAR-100 datasets.

Ramtin Madani, Mersedeh Ashraphijuo, Mohsen Kheirandishfard et al.

In the first part of this work [32], we introduce a convex parabolic relaxation for quadratically-constrained quadratic programs, along with a sequential penalized parabolic relaxation algorithm to recover near-optimal feasible solutions. In this second part, we show that starting from a feasible solution or a near-feasible solution satisfying certain regularity conditions, the sequential penalized parabolic relaxation algorithm convergences to a point which satisfies Karush-Kuhn-Tucker optimality conditions. Next, we present numerical experiments on benchmark non-convex QCQP problems as well as large-scale instances of system identification problem demonstrating the efficiency of the proposed approach.

Ramtin Madani, Mersedeh Ashraphijuo, Mohsen Kheirandishfard et al.

For general quadratically-constrained quadratic programming (QCQP), we propose a parabolic relaxation described with convex quadratic constraints. An interesting property of the parabolic relaxation is that the original non-convex feasible set is contained on the boundary of the parabolic relaxation. Under certain assumptions, this property enables one to recover near-optimal feasible points via objective penalization. Moreover, through an appropriate change of coordinates that requires a one-time computation of an optimal basis, the easier-to-solve parabolic relaxation can be made as strong as a semidefinite programming (SDP) relaxation, which can be effective in accelerating algorithms that require solving a sequence of convex surrogates. The majority of theoretical and computational results are given in the next part of this work [57].

Anna Deza, Andrés Gómez, Alper Atamtürk

This paper introduces mixed-integer optimization methods to solve regression problems that incorporate fairness metrics. We propose an exact formulation for training fair regression models. To tackle this computationally hard problem, we study the polynomially-solvable single-factor and single-observation subproblems as building blocks and derive their closed convex hull descriptions. Strong formulations obtained for the general fair regression problem in this manner are utilized to solve the problem with a branch-and-bound algorithm exactly or as a relaxation to produce fair and accurate models rapidly. Moreover, to handle large-scale instances, we develop a coordinate descent algorithm motivated by the convex-hull representation of the single-factor fair regression problem to improve a given solution efficiently. Numerical experiments conducted on fair least squares and fair logistic regression problems show competitive statistical performance with state-of-the-art methods while significantly reducing training times.

Anna Deza, Alper Atamturk

In logistic regression, it is often desirable to utilize regularization to promote sparse solutions, particularly for problems with a large number of features compared to available labels. In this paper, we present screening rules that safely remove features from logistic regression with $\ell_0-\ell_2$ regularization before solving the problem. The proposed safe screening rules are based on lower bounds from the Fenchel dual of strong conic relaxations of the logistic regression problem. Numerical experiments with real and synthetic data suggest that a high percentage of the features can be effectively and safely removed apriori, leading to substantial speed-up in the computations.

Linchuan Wei, Alper Atamtürk, Andrés Gómez et al.

We consider the convex quadratic optimization problem with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic number of additional variables consists of a single positive semidefinite constraint (explicitly stated) and linear constraints. In particular, convexification of this class of problems reduces to describing a polyhedral set in an extended formulation. While the vertex representation of this polyhedral set is exponential and an explicit linear inequality description may not be readily available in general, we derive a compact mixed-integer linear formulation whose solutions coincide with the vertices of the polyhedral set. We also give descriptions in the original space of variables: we provide a description based on an infinite number of conic-quadratic inequalities, which are ``finitely generated." In particular, it is possible to characterize whether a given inequality is necessary to describe the convex hull. The new theory presented here unifies several previously established results, and paves the way toward utilizing polyhedral methods to analyze the convex hull of mixed-integer nonlinear sets.

Alper Atamturk, Andres Gomez

We study the minimization of a rank-one quadratic with indicators and show that the underlying set function obtained by projecting out the continuous variables is supermodular. Although supermodular minimization is, in general, difficult, the specific set function for the rank-one quadratic can be minimized in linear time. We show that the convex hull of the epigraph of the quadratic can be obtaining from inequalities for the underlying supermodular set function by lifting them into nonlinear inequalities in the original space of variables. Explicit forms of the convex-hull description are given, both in the original space of variables and in an extended formulation via conic quadratic-representable inequalities, along with a polynomial separation algorithm. Computational experiments indicate that the lifted supermodular inequalities in conic quadratic form are quite effective in reducing the integrality gap for quadratic optimization with indicators.

Alper Atamtürk, Andrés Gómez

We give safe screening rules to eliminate variables from regression with $\ell_0$ regularization or cardinality constraint. These rules are based on guarantees that a feature may or may not be selected in an optimal solution. The screening rules can be computed from a convex relaxation solution in linear time, without solving the $\ell_0$ optimization problem. Thus, they can be used in a preprocessing step to safely remove variables from consideration apriori. Numerical experiments on real and synthetic data indicate that, on average, 76\% of the variables can be fixed to their optimal values, hence, reducing the computational burden for optimization substantially. Therefore, the proposed fast and effective screening rules extend the scope of algorithms for $\ell_0$-regression to larger data sets.

Alper Atamturk, Vishnu Narayanan

Using polarity, we give an outer polyhedral approximation for the epigraph of set functions. For a submodular function, we prove that the corresponding polar relaxation is exact; hence, it is equivalent to the Lovász extension. The polar approach provides an alternative proof for the convex hull description of the epigraph of a submodular function. Computational experiments show that the inequalities from outer approximations can be effective as cutting planes for solving submodular as well as non-submodular set function minimization problems.

Alper Atamturk, Andres Gomez

Sparse regression models are increasingly prevalent due to their ease of interpretability and superior out-of-sample performance. However, the exact model of sparse regression with an $\ell_0$ constraint restricting the support of the estimators is a challenging (\NP-hard) non-convex optimization problem. In this paper, we derive new strong convex relaxations for sparse regression. These relaxations are based on the ideal (convex-hull) formulations for rank-one quadratic terms with indicator variables. The new relaxations can be formulated as semidefinite optimization problems in an extended space and are stronger and more general than the state-of-the-art formulations, including the perspective reformulation and formulations with the reverse Huber penalty and the minimax concave penalty functions. Furthermore, the proposed rank-one strengthening can be interpreted as a \textit{non-separable, non-convex, unbiased} sparsity-inducing regularizer, which dynamically adjusts its penalty according to the shape of the error function without inducing bias for the sparse solutions. In our computational experiments with benchmark datasets, the proposed conic formulations are solved within seconds and result in near-optimal solutions (with 0.4\% optimality gap) for non-convex $\ell_0$-problems. Moreover, the resulting estimators also outperform alternative convex approaches from a statistical perspective, achieving high prediction accuracy and good interpretability.

Alper Atamturk, Andres Gomez, Shaoning Han

Signal estimation problems with smoothness and sparsity priors can be naturally modeled as quadratic optimization with $\ell_0$-"norm" constraints. Since such problems are non-convex and hard-to-solve, the standard approach is, instead, to tackle their convex surrogates based on $\ell_1$-norm relaxations. In this paper, we propose a new iterative (convex) conic quadratic relaxations that exploit not only the $\ell_0$-"norm" terms, but also the fitness and smoothness functions. The iterative convexification approach substantially closes the gap between the $\ell_0$-"norm" and its $\ell_1$ surrogate. These stronger relaxations lead to significantly better estimators than $\ell_1$-norm approaches and also allow one to utilize affine sparsity priors. In addition, the parameters of the model and the resulting estimators are easily interpretable. Experiments with a tailored Lagrangian decomposition method indicate that the proposed iterative convex relaxations \rev{yield solutions within 1\% of the exact $\ell_0$ approach, and can tackle instances with up to 100,000 variables under one minute.