**Dr. Milagros Loreto, Faculty Mentor**

Optimization is always an exciting subject for research. In particular, emerging methods such as Modified Spectral Projected Subgradient (MSPS) are suitable topics for undergraduate research since they tackle nonsmooth optimization problems, arising from modeling of real life situations. Dr. Loreto has mentored 4 undergraduate research works focused on these methods, one of them was a poster presentation. Dr. Loreto has authored two articles on this topic. After learning a few concepts and getting familiar with the codebase students will be able to participate in the next research projects.

• Conditional Modified Spectral Projected Subgradient (CMSPS). The phenomenon of zigzagging is present on subgradient algorithms. The zigzagging phenomenon of kind II is caused by subgradients almost perpendicular to the face of the feasible set {(λ1, λ2) : λ1 = λ2} and it could be evident on iterations of the (MSPS) method for some problems. The main objective of this project is to identify and correct the zigzagging phenomenon of kind II for MSPS. Students will develop changes on the original MSPS algorithm to detect and to attenuate the zigzagging, conduce numerical experimentation using Matlab and compare the resulting algorithm to the original MSPS. During the numerical experimentation, students will get familiar with larger problems coming from the OR-Library while they learn how to use the MSPS to solve these problems.

• Incremental Modified Spectral Projected Subgradient (IMSPS). The MSPS was created as an alternative to solve the lagrangean dual problem assoaciated to an integer problem, with low storage and computational cost. In order to adapt the MSPS to other kind of non-differentiable problems, the minimization of a function that consists of the sum of a large number of component functions will be considered. The students will develop changes on the MSPS following the incremental techniques proposed in to solve this kind of minimization problem. Also, the students will perform numerical experimentation on the examples proposed in the same work.