Several summer interns presented their work related to MACS.

**On Bilevel Mixed-Integer Nonlinear Programming Problems**

1:15 p.m. – 1:30 p.m.

*Presenter*: Harikrishnan Sreekumaran (Supervisor: Sven Leyffer)

*Abstract*: Bilevel programming provides a natural framework for modelling hierarchical decision making, with applications in varied fields such as market design, network planning and vulnerability analysis, and engineering design among others. In this talk, we review bilevel mixed integer nonlinear programming problems (BLMINLPs) and describe the significant challenges in the analysis and algorithm design for such problems. We present relaxation formulations for certain classes of BLMINLPs and introduce algorithms for solving such problems using these relaxations. Finally we present a few application examples and some preliminary numerical results.

**Building an Adjoint Based Dynamics Constrained Optimization of Electrical Power Systems**

1:30 p.m. – 1:45 p.m.

*Presenter*: Paul Tranquilli (Supervisor: Cosmin Petra)

*Slides of Paul*

*Abstract*: We present a mathematical framework for the solution of optimal power flow, including dynamic security constraints, using adjoint based sensitivities. We briefly discuss some implementation details of using PETSc to perform both the forward and backward time integrations.

**Automatic Discretization of ODE and PDE Systems Using Pyomo**

1:45 p.m.- 2:00 p.m.

*Presenter*: Bethany Nicholson (Supervisor: Victor Zavala)

*Slides of Bethany*

*Abstract*: Dynamic optimization problems that include ordinary/partial differential equations as constraints are typically solved by discretizing the model and then solving the resulting nonlinear problem. This talk introduces a way to do model discretization automatically using the open-source modeling language Pyomo. We will discuss the straight-forward syntax behind this new functionality as well as the flexible discretization options that are available. Finally, we will look at a couple sample problems that show the wide range of dynamic optimzation problems that can now be represented and solved using Pyomo.

**Derivative-based Solution of the Constrained Optimization Problem(s) in DeMarco’s Model**

2:00 p.m. – 2:15 p.m.

*Presenter*: Ahmed Attia (Supervisor: Mihai Anitescu)

*Slides of Ahmed*

*Abstract*: Predicting cascading network failure is a vital problem for the ever increasing scale of engineered systems as electric power grids, communication networks, and internet. To predict cascading failures, for one or more branches, the bottleneck is a constrained optimization step. Without derivative information, this optimization step takes a very long time even for the simplest settings, and may even fail to converge for large simulation times. We considered a simple, but general, model developed by DeMarco to understand how small-scale failures of individual elements may propagate to produce global failures. My main task was to derive and implement the adjoint of the quadratic constraint(s). The derivative information of the cost functional and constraints, made the optimization very fast (roughly 100-150 times faster). The results are being extended now to large-scale settings.

**Bound Contraction Algorithm for Global Optimization of Natural Gas Networks**

2:15 p.m. – 2:30 p.m.

*Presenter*: Francisco Trespalacios-Sagues (Supervisor: Victor Zavala)

*Abstract*: The natural gas transportation network problem has received increased attention in recent years. In this work, we first derive a convex NLP for this problem, based on a well-known non-convex NLP and two operating assumptions. We then present a non-convex model that includes the cost of both: purchasing natural gas from suppliers, and operating compressors in the network. Finally, we develop a bound contraction algorithm to solve this non-convex model. The algorithm is tested in a simplified real network, and compared against state of the art global solvers.