3 Presentations

• 
  08:45 AM - 09:10 AM

  Quadratic regularization with cubic descent for unconstrained optimization

  • Ernesto G. Birgin, presenter, University of São Paulo
  • J. M. Martínez, University of Campinas

  Cubic-regularization and trust-region methods with worst-case first-order complexity $O(\varepsilon^{-3/2})$ and worst-case second-order complexity $O(\varepsilon^{-3})$ have been developed in the last few years. In this paper it is proved that the same complexities are achieved by means of a quadratic-regularization method with a cubic sufficient-descent condition instead of the more usual predicted-reduction based descent. Asymptotic convergence and order of convergence results are also presented. Finally, some numerical experiments comparing the new algorithm with a well-established quadratic regularization method are shown.

• 
  09:10 AM - 09:35 AM

  Robust optimization for nonlinear process design and operations problem

  • Zukui Li, presenter, University of Alberta

  A novel robust optimization framework is proposed to address general nonlinear optimization problems under uncertainty in process design and operations. The proposed framework can deal with problems with equality constraints associated with uncertainty. Local linearization is made in respect to the uncertain parameters around multiple realizations of the uncertainty, and an iterative algorithm is implemented to solve the robust optimization problem. Local linear decision rule is adopted for the adjustable operation variables. Different applications with various levels of complexity are used to demonstrate the effectiveness of the proposed nonlinear robust optimization framework.

• 
  09:35 AM - 10:00 AM

  A Customized Branch-and-Bound Approach for Irregular Shape Nesting

  • Akang Wang, Carnegie Mellon University
  • Hanselman Christopher L., Carnegie Mellon University
  • Gounaris Chrysanthos E., presenter, Carnegie Mellon University

  We study the problem of determining a non-overlapping placement of a given set of two-dimensional shapes such that the shapes can fit in an as small as possible enclosing box. This nesting problem has ubiquitous applications in a number of industries where it is desirable to minimize the waste of a two-dimensional material resource from which a number of smaller articles need to be carved. When the shapes are irregular, such as when the shapes are non-convex or feature interior holes, obtaining a provably optimal solution becomes very challenging. The traditional approach calls for approximating the original shapes via a set of inscribed circles and enforcing the shape-shape non-overlap restrictions via reverse convex quadratic constraints, leading to a global optimization problem. In this paper, we develop a problem-specific linear programming relaxation that can be arbitrarily tightened via judiciously chosen branching decisions in the context of a branch-and-bound search process. Computational experiments on a suite of literature benchmarks demonstrate that our method can outperform the traditional approach even when state-of-the-art global optimization solvers are employed.