14 results on '"Ribeiro, Ademir A."'
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2. How to correctly answer 'Is the optimal critical point a local minimizer?' in Calculus courses.
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Ribeiro, Ademir Alves and Barbosa, José Renato Ramos
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CALCULUS education , *CRITICAL point theory , *MULTIVARIATE analysis , *LAGRANGE multiplier ,UNDERGRADUATE education - Abstract
This short note discusses how the optimality conditions for minimizing a multivariate function subject to equality constraints have been covered in some undergraduate Calculus courses. In particular, we will focus on the most common optimization problems in Calculus of several variables: the 2 and 3-dimensional cases. So, along with sufficient conditions for a critical point to be a local minimizer, we also present and discuss counterexamples for some statements that can be found in the literature of undergraduate Calculus related to Lagrange Multipliers, such as 'between the critical points, the ones which have the smallest image (under the function) are minimizers' or 'a single critical point (which is a local minimizer) is a global minimizer'. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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3. A Comparative Study of Sequential Optimality Conditions for Mathematical Programs with Cardinality Constraints.
- Author
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Ribeiro, Ademir A., Sachine, Mael, and Krulikovski, Evelin H. M.
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NONLINEAR programming , *CLASSICAL conditioning , *COMPARATIVE studies - Abstract
We propose a comparative study of sequential optimality conditions for mathematical programs with cardinality constraints. Besides analyzing some of the classical approximate conditions for nonlinear programming, such as AKKT, CAKKT and PAKKT, we also propose an approximate weak stationarity ( AW -stationarity) concept designed to deal with this class of problems and we prove that it is a legitimate optimality condition independently of any constraint qualification. We point out that, despite the computational appeal of the sequential optimality conditions, in this work we are not concerned with algorithmic consequences. Our aim is purely to discuss theoretical aspects of such conditions for MPCaC problems. [ABSTRACT FROM AUTHOR]
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- 2022
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4. On the optimal separating hyperplane for arbitrary sets: a generalization of the SVM formulation and a convex hull approach.
- Author
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Ribeiro, Ademir A. and Sachine, Mael
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SUPPORT vector machines , *GENERALIZATION - Abstract
We generalize the existing formulation and results on linear separability of sets. In order to characterize the solution of the generalized problem, we use the concepts of convex hulls. For finite sets, it is well known the Support Vector Machine technique for finding the optimal separating hyperplane. Here we consider arbitrary sets, allowing infinite, unbounded and nonclosed sets. The problem is formulated as an optimization problem with possibly infinitely many constraints. We prove existence and uniqueness of the solution. Besides, we present some examples and counterexamples to many properties discussed in the text and statements in the literature. [ABSTRACT FROM AUTHOR]
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- 2022
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5. On the Weak Stationarity Conditions for Mathematical Programs with Cardinality Constraints: A Unified Approach.
- Author
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Krulikovski, Evelin H. M., Ribeiro, Ademir A., and Sachine, Mael
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NONLINEAR programming , *MOTIVATION (Psychology) - Abstract
In this paper, we study a class of optimization problems, called Mathematical Programs with Cardinality Constraints (MPCaC). This kind of problem is generally difficult to deal with, because it involves a constraint that is not continuous neither convex, but provides sparse solutions. Thereby we reformulate MPCaC in a suitable way, by modeling it as mixed-integer problem and then addressing its continuous counterpart, which will be referred to as relaxed problem. We investigate the relaxed problem by analyzing the classical constraints in two cases: linear and nonlinear. In the linear case, we propose a general approach and present a discussion of the Guignard and Abadie constraint qualifications, proving in this case that every minimizer of the relaxed problem satisfies the Karush–Kuhn–Tucker (KKT) conditions. On the other hand, in the nonlinear case, we show that some standard constraint qualifications may be violated. Therefore, we cannot assert about KKT points. Motivated to find a minimizer for the MPCaC problem, we define new and weaker stationarity conditions, by proposing a unified approach. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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6. The complexity of primal-dual fixed point methods for ridge regression.
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Ribeiro, Ademir Alves and Richtárik, Peter
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FIXED point theory , *REGRESSION analysis , *MATHEMATICAL regularization , *PARAMETERS (Statistics) , *MATHEMATICAL bounds , *LINEAR systems - Abstract
We study the ridge regression ( L 2 regularized least squares) problem and its dual, which is also a ridge regression problem. We observe that the optimality conditions describing the primal and dual optimal solutions can be formulated in several different but equivalent ways. The optimality conditions we identify form a linear system involving a structured matrix depending on a single relaxation parameter which we introduce for regularization purposes. This leads to the idea of studying and comparing, in theory and practice, the performance of the fixed point method applied to these reformulations. We compute the optimal relaxation parameters and uncover interesting connections between the complexity bounds of the variants of the fixed point scheme we consider. These connections follow from a close link between the spectral properties of the associated matrices. For instance, some reformulations involve purely imaginary eigenvalues; some involve real eigenvalues and others have all eigenvalues on the complex circle. We show that the deterministic Quartz method—which is a special case of the randomized dual coordinate ascent method with arbitrary sampling recently developed by Qu, Richtárik and Zhang—can be cast in our framework, and achieves the best rate in theory and in numerical experiments among the fixed point methods we study. Remarkably, the method achieves an accelerated convergence rate. Numerical experiments indicate that our main algorithm is competitive with the conjugate gradient method. [ABSTRACT FROM AUTHOR]
- Published
- 2018
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7. Global convergence of a general filter algorithm based on an efficiency condition of the step.
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Periçaro, Gislaine A., Ribeiro, Ademir A., and Karas, Elizabeth W.
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STOCHASTIC convergence , *COMPUTER algorithms , *FEASIBILITY studies , *MATHEMATICAL functions , *PROOF theory , *PERFORMANCE evaluation , *CRITERION (Theory of knowledge) - Abstract
Abstract: In this work we discuss global convergence of a general filter algorithm that depends neither on the definition of the forbidden region, which can be given by the original or slanting filter rule, nor on the way in which the step is computed. This algorithm basically consists of calculating a point not forbidden by the filter from the current point. Assuming that this step must be efficient, in the sense that near a feasible non-stationary point the decrease in the objective function is relatively large, we prove the global convergence of the algorithm. We also discuss that such a condition is satisfied if the step is computed by the SQP or Inexact Restoration methods. For SQP we present a general proof of this result that is valid for both the original and the slanting filter criterion. In order to compare the performance of the general filter algorithm according to the method used to calculate the step and the filter rule regarded, we present numerical experiments performed with problems from CUTEr collection. [Copyright &y& Elsevier]
- Published
- 2013
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8. A bundle-filter method for nonsmooth convex constrained optimization.
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Karas, Elizabeth, Ribeiro, Ademir, Sagastizábal, Claudia, and Solodov, Mikhail
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CONSTRAINED optimization , *NONSMOOTH optimization , *MATHEMATICAL optimization , *FILTERS (Mathematics) , *ALGORITHMS , *MATHEMATICAL programming - Abstract
For solving nonsmooth convex constrained optimization problems, we propose an algorithm which combines the ideas of the proximal bundle methods with the filter strategy for evaluating candidate points. The resulting algorithm inherits some attractive features from both approaches. On the one hand, it allows effective control of the size of quadratic programming subproblems via the compression and aggregation techniques of proximal bundle methods. On the other hand, the filter criterion for accepting a candidate point as the new iterate is sometimes easier to satisfy than the usual descent condition in bundle methods. Some encouraging preliminary computational results are also reported. [ABSTRACT FROM AUTHOR]
- Published
- 2009
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9. GLOBAL CONVERGENCE OF FILTER METHODS FOR NONLINEAR PROGRAMMING.
- Author
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RIBEIRO, ADEMIR A., KARAS, ELIZABETH W., and GONZAGA, CLÓVIS C.
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ALGORITHMS , *NONLINEAR programming , *STOCHASTIC convergence , *ASYMPTOTIC expansions , *QUADRATIC programming - Abstract
We present a general filter algorithm that allows a great deal of freedom in the step computation. Each iteration of the algorithm consists basically in computing a point which is not forbidden by the filter, from the current point. We prove its global convergence, assuming that the step must be efficient, in the sense that, near a feasible nonstationary point, the reduction of the objective function is "large." We show that this condition is reasonable, by presenting two classical ways of performing the step which satisfy it. In the first one, the step is obtained by the inexact restoration method of Martínez and Pilotta. In the second, the step is computed by sequential quadratic programming. [ABSTRACT FROM AUTHOR]
- Published
- 2008
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10. Theoretical analysis of classic and capacity constrained fuzzy clustering.
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Benatti, Kléber A., Pedroso, Lucas G., and Ribeiro, Ademir A.
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POINT set theory , *A priori - Abstract
In this paper we present a theoretical analysis on fuzzy centroid-based clustering methods. In addition to the formulation on the classical approaches, we consider constraints that may be useful in some practical applications, such as restrictions on the number of points in each group, and methods that deal with these constraints. We propose a more general formulation to the constrained clustering problem, where each point has an associated weight, and the sum of the weights of the points that compose each group is established a priori. For both classical and proposed approaches we discuss existence and uniqueness of solutions of the involved problems, providing mathematical foundations for the established formulas. Preliminary numerical experiments, performed by means of two-dimensional examples, are also presented. [ABSTRACT FROM AUTHOR]
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- 2022
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11. On the convergence of augmented Lagrangian strategies for nonlinear programming.
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Andreani, Roberto, Ramos, Alberto, Ribeiro, Ademir A, Secchin, Leonardo D, and Velazco, Ariel R
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NONLINEAR programming , *ALGORITHMS , *CONSTRAINED optimization - Abstract
Augmented Lagrangian (AL) algorithms are very popular and successful methods for solving constrained optimization problems. Recently, global convergence analysis of these methods has been dramatically improved by using the notion of sequential optimality conditions. Such conditions are necessary for optimality, regardless of the fulfillment of any constraint qualifications, and provide theoretical tools to justify stopping criteria of several numerical optimization methods. Here, we introduce a new sequential optimality condition stronger than previously stated in the literature. We show that a well-established safeguarded Powell–Hestenes–Rockafellar (PHR) AL algorithm generates points that satisfy the new condition under a Lojasiewicz-type assumption, improving and unifying all the previous convergence results. Furthermore, we introduce a new primal–dual AL method capable of achieving such points without the Lojasiewicz hypothesis. We then propose a hybrid method in which the new strategy acts to help the safeguarded PHR method when it tends to fail. We show by preliminary numerical tests that all the problems already successfully solved by the safeguarded PHR method remain unchanged, while others where the PHR method failed are now solved with an acceptable additional computational cost. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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12. Fenchel–Moreau conjugation for lower semi-continuous functions.
- Author
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Cotrina, John, Karas, Elizabeth W., Ribeiro, Ademir A., Sosa, Wilfredo, and Yuan, Jin Y.
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CONJUGATE gradient methods , *CONTINUOUS functions , *CONVEX functions , *DUALITY theory (Mathematics) , *TOPOLOGICAL spaces , *MATHEMATICAL optimization - Abstract
We introduce a modification of Fenchel's conjugation which is a particular case of Moreau's conjugation. We obtain good properties such as convexity of the conjugate function even though the function is not convex. We also introduce the concept of conjugate dual space as a class of continuous operators, while in the Fenchel conjugation the conjugate dual space is the classical topological dual space. Finally, we present some examples for illustrating the difference between the Fenchel–Moreau conjugation and our modification. [ABSTRACT FROM PUBLISHER]
- Published
- 2011
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13. Local convergence of filter methods for equality constrained non-linear programming.
- Author
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Karas, Elizabeth W., Gonzaga, Clóvis C., and Ribeiro, Ademir A.
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STOCHASTIC convergence , *ALGORITHMS , *NONLINEAR programming , *LINEAR substitutions , *QUADRATIC programming - Abstract
In Gonzaga et al. [A globally convergent filter method for nonlinear programming, SIAM J. Optimiz. 14 (2003), pp. 646-669] we discuss general conditions to ensure global convergence of inexact restoration filter algorithms for non-linear programming. In this article we show how to avoid the Maratos effect by means of a second-order correction. The algorithms are based on feasibility and optimality phases, which can be either independent or not. The optimality phase differs from the original one only when a full Newton step for the tangential minimization of the Lagrangian is efficient but not acceptable by the filter method. In this case a second-order corrector step tries to produce an acceptable point keeping the efficiency of the rejected step. The resulting point is tested by trust region criteria. Under the usual hypotheses, the algorithm inherits the quadratic convergence properties of the feasibility and optimality phases. This article includes a comparison between classical Sequential Quadratic Programming (SQP) and Inexact Restoration (IR) iterations, showing that both methods share the same asymptotic convergence properties. [ABSTRACT FROM AUTHOR]
- Published
- 2010
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14. Global convergence of slanting filter methods for nonlinear programming
- Author
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Karas, Elizabeth W., Oening, Ana P., and Ribeiro, Ademir A.
- Subjects
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NONLINEAR programming , *ALGORITHMS , *MATHEMATICAL optimization , *MATHEMATICAL analysis - Abstract
Abstract: In this paper, we present a general algorithm for nonlinear programming which uses a slanting filter criterion for accepting the new iterates. Independently of how these iterates are computed, we prove that all accumulation points of the sequence generated by the algorithm are feasible. Computing the new iterates by the inexact restoration method, we prove stationarity of all accumulation points of the sequence. [Copyright &y& Elsevier]
- Published
- 2008
- Full Text
- View/download PDF
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