In this paper, we deal with the Front Steepest Descent algorithm for multi-objective optimization. We point out that the algorithm from the literature is often incapable, by design, of spanning large portions of the Pareto front. We thus introduce some modifications within the algorithm aimed to overcome this significant limitation. We prove that the asymptotic … Read more
Logistic regression is one of the most popular methods in binary classification, wherein estimation of model parameters is carried out by solving the maximum likelihood (ML) optimization problem, and the ML estimator is defined to be the optimal solution of this problem. It is well known that the ML estimator exists when the data is … Read more
A number of first-order methods have been proposed for smooth multiobjective optimization for which some form of convergence to first order criticality has been proved. Such convergence is global in the sense of being independent of the starting point. In this paper we analyze the rate of convergence of gradient descent for smooth unconstrained multiobjective … Read more
We consider the gradient (or steepest) descent method with exact line search applied to a strongly convex function with Lipschitz continuous gradient. We establish the exact worst-case rate of convergence of this scheme, and show that this worst-case behavior is exhibited by a certain convex quadratic function. We also extend the result to a noisy … Read more
In this paper we analyze boosting algorithms in linear regression from a new perspective: that of modern first-order methods in convex optimization. We show that classic boosting algorithms in linear regression, namely the incremental forward stagewise algorithm (FS-epsilon) and least squares boosting (LS-Boost-epsilon), can be viewed as subgradient descent to minimize the loss function defined … Read more
Motivated by Bland’s linear-programming generalization of the renowned Edmonds-Karp efficient refinement of the Ford-Fulkerson maximum-flow algorithm, we discuss three closely-related natural augmentation rules for linear and integer-linear optimization. In several nice situations, we show that polynomially-many augmentation steps suffice to reach an optimum. In particular, when using “discrete steepest-descent augmentations” (i.e., directions with the best … Read more
The worst-case complexity of the steepest-descent algorithm with exact linesearches for unconstrained smooth optimization is analyzed, and it is shown that the number of iterations of this algorithm which may be necessary to find an iterate at which the norm of the objective function’s gradient is less that a prescribed $\epsilon$ is, essentially, a multiple … Read more
In this paper, we consider a class of inexact proximal point methods for convex optimization which allows a relative error tolerance in the approximate solution of each proximal subproblem. By exploiting the special structure of convex optimization problems, we are able to derive surprising complexity bounds for the aforementioned class. As a consequence, we show … Read more
The possibilities inherent in steepest descent methods have been considerably amplified by the introduction of the Barzilai-Borwein choice of step-size, and other related ideas. These methods have proved to be competitive with conjugate gradient methods for the minimization of large dimension unconstrained minimization problems. This paper suggests a method which is able to take advantage … Read more
It is shown that the steepest descent and Newton’s method for unconstrained nonconvex optimization under standard assumptions may be both require a number of iterations and function evaluations arbitrarily close to O(epsilon^{-2}) to drive the norm of the gradient below epsilon. This shows that the upper bound of O(epsilon^{-2}) evaluations known for the steepest descent … Read more