Abstract : Continuous optimization is one of the main areas in Applied Mathematics. It has
plentiful applications and rich theory and algorithms. This mini-symposium
presents recent advances in the area, starting from important theoretical
aspects of optimality conditions that guide the development of different
algorithms based on higher order information, like Newton-type and third-order
algorithms. Many of the methods also exploit the specific structure of the
underlying application to achieve high performance. Finally, the mini-symposium
also pays tribute to José Mario Martínez’s influence in the field. The
contributions can be somewhat linked to his insightful ideas in different
periods of his career.
Organizer(s) : Paulo J. S. Silva, Roberto Andreani
[03840] Constant rank constraint qualification for nonlinear second-order cone programming
Format : Talk at Waseda University
Author(s) :
Gabriel Haeser (University of Sao Paulo)
Abstract : We revisit the classical notions of nondegeneracy and Robinson's condition in the context of nonlinear second-order cone programming. For an m-dimensional second-order cone, instead of stating nondegeneracy at the vertex as the linear independence of m derivative vectors, we do it in terms of several statements of linear independence of two derivative vectors. This allows embedding the structure of the second-order cone into the formulation of the conditions, providing weaker variants and applications.
[03851] Strong global convergence properties of an Augmented Lagrangian method for symmetric cones
Format : Talk at Waseda University
Author(s) :
Daiana Oliveira dos Santos (UNIFESP)
Abstract : Sequential optimality conditions have played a major role in proving stronger global convergence results for numerical algorithms used in nonlinear programming. Several extensions have been described in conic contexts, leading to many open questions. In this talk, we will present new sequential optimality conditions for nonlinear symmetric cone programming. Stronger results are obtained by exploiting the rich algebraic structure of the problem.
[03586] Adaptive Third-Order Methods for Composite Convex Optimization
Format : Talk at Waseda University
Author(s) :
Geovani Grapiglia (UCLouvain)
Yurii Nesterov (UCLouvain)
Abstract : In this talk we present adaptive third-order methods for composite convex optimization problems in which the smooth part has Lipschitz continuous third-order derivatives. In our new schemes the regularization parameters are tunned by checking the progress of the inner solver used to compute trial points. In particular, this technique allows us to design an adaptive accelerated method that can find an $\epsilon$-approximate solution using at most $\mathcal{O}\left(|\log(\epsilon)|\epsilon^{-\frac{1}{4}}\right)$ iterations of the inner solver.
[03706] On the globalization of nonlinear programming methods.
Format : Talk at Waseda University
Author(s) :
L. Felipe Bueno (Universidade Federal de São Paulo)
Abstract : In this work we identify some very general conditions that guarantee the global convergence of numerical methods to solve constrained nonlinear optimization problems. With that in mind, we show that a range of Inexact Restoration methods fit the presented framework. Furthermore, we propose a particular algorithm in this line, combined with an acceleration process supported by the general theory of convergence. Computational tests attest to the efficiency of the proposed strategy.
[04137] Proportionality based algorithms for quadratic programming
Format : Talk at Waseda University
Author(s) :
Gerardo Toraldo (Università della Campania "L.Vanvitelli")
William W. Hager (University of Florida)
Marco Viola (University College Dublin)
Abstract : We present a decomposition of the negative gradient on the tangent cone at a feasible point for optimization problems with polyhedral constraints. This decomposition, based on the idea of proportioning, extends the definition of the free and chopped gradient in bound constrained optimization. Such decomposition allows us to generalize the definition of binding variables and to measure complementary aspects of stationarity that can be exploited to design effective switching rules in Gradient Projection two-phase algorithms.
[02990] On enhanced KKT optimality conditions for smooth nonlinear optimization
Format : Talk at Waseda University
Author(s) :
Roberto Andreani (State University of Campinas/ UNICAMP)
Abstract : The Fritz-John and Karush-Kuhn-Tucker (KKT) conditions are crucial for
finding minimizers in constrained optimization. They have been
augmented with extra necessary conditions since the 1970s, with the
enhanced KKT stationarity being one of them. This work focuses on
enhanced KKT stationarity for smooth nonlinear programming and
analyzes improved multipliers with quasi-normality. The results have
implications for sequential optimality conditions and complementarity
constraints in multi-objective problems.
[03943] On the Inexact Restoration approach for adaptive sample size in finite sum minimization
Format : Talk at Waseda University
Author(s) :
Stefania Bellavia (University of Florence)
Natasa Krejic (University of Novi Sad)
Benedetta Morini (University of Florence)
Simone Rebegoldi (University of Florence)
Abstract : In this talk we discuss recent advances in the inexact restoration approach combined with stochastic trust-region methods for finite-sum minimization problems. At each iteration, the proposed methods approximate the function and the derivatives by subsampling. The choice of the function sample size is ruled by the Inexact Restoration approach, whereas the derivatives approximations are computed averaging in smaller sets.
We report worst-case complexity results in expectation and numerical results showing the advantages of adaptive approaches.
[03290] Block coordinate descent and the close enough traveling salesman problem
Format : Talk at Waseda University
Author(s) :
Ernesto G. Birgin (University of São Paulo)
Abstract : At each iteration of a Block Coordinate Descent method one minimizes a constrained approximation of the objective function with respect to a generally small set of variables. In this work we address the problem in which block constraints are not defined by global sets of equations and inequations. An algorithm is defined and convergence and complexity are proved. The proposed method is used to solve a generalization of the close enough traveling salesman problem.
[02948] Sequential optimality conditions: how to stop optimization algorithms
Format : Talk at Waseda University
Author(s) :
Paulo J. S. Silva (University of Campinas)
Abstract : Optimality conditions, like KKT, play essential roles in modern optimization. They may be used as a starting point to develop algorithms or as a condition to accept an approximate solution. This dynamic aspect, led to the development of sequential optimality conditions that try to capture the iterative approximation nature of computed sequences. In this talk, we will present how this development enlightened the convergence requirements and termination criteria of algorithms in nonlinear optimization.
