Omid Nohadani

Robust Optimization for Unconstrained Simulation-Based Problems

In engineering design, an optimized solution often turns out to be suboptimal when errors are encountered. Although the theory of robust convex optimization has taken significant strides over the past decade, all approaches fail if the underlying cost function is not explicitly given; it is even worse if the cost function is nonconvex. In this work, we present a robust optimization method that is suited for unconstrained problems with a nonconvex cost function as well as for problems based on simulations, such as large partial differential equations (PDE) solver, response surface, and Kriging metamodels. Moreover, this technique can be employed for most real-world problems because it operates directly on the response surface and does not assume any specific structure of the problem. We present this algorithm along with the application to an actual engineering problem in electromagnetic multiple scattering of aperiodically arranged dielectrics, relevant to nanophotonic design. The corresponding objective function is highly nonconvex and resides in a 100-dimensional design space. Starting from an “optimized” design, we report a robust solution with a significantly lower worst-case cost, while maintaining optimality. We further generalize this algorithm to address a nonconvex optimization problem under both implementation errors and parameter uncertainties.

Robust optimization with simulated annealing

Complex systems can be optimized to improve the performance with respect to desired functionalities. An optimized solution, however, can become suboptimal or even infeasible, when errors in implementation or input data are encountered. We report on a robust simulated annealing algorithm that does not require any knowledge of the problems structure. This is necessary in many engineering applications where solutions are often not explicitly known and have to be obtained by numerical simulations. While this nonconvex and global optimization method improves the performance as well as the robustness, it also warrants for a global optimum which is robust against data and implementation uncertainties. We demonstrate it on a polynomial optimization problem and on a high-dimensional and complex nanophotonic engineering problem and show significant improvements in efficiency as well as in actual optimality.

A hybrid approach to beam angle optimization in intensity-modulated radiation therapy

Intensity-Modulated Radiation Therapy is the technique of delivering radiation to cancer patients by using non-uniform radiation fields from selected angles, with the aim of reducing the intensity of the beams that go through critical structures while reaching the dose prescription in the target volume. Two decisions are of fundamental importance: to select the beam angles and to compute the intensity of the beams used to deliver the radiation to the patient. Often, these two decisions are made separately: first, the treatment planners, on the basis of experience and intuition, decide the orientation of the beams and then the intensities of the beams are optimized by using an automated software tool. Automatic beam angle selection (also known as Beam Angle Optimization) is an important problem and is today often based on human experience. In this context, we face the problem of optimizing both the decisions, developing an algorithm which automatically selects the beam angles and computes the beam intensities. We propose a hybrid heuristic method, which combines a simulated annealing procedure with the knowledge of the gradient. Gradient information is used to quickly find a local minimum, while simulated annealing allows to search for global minima. As an integral part of this procedure, the beam intensities are optimized by solving a Linear Programming model. The proposed method presents a main difference from previous works: it does not require to have on input a set of candidate beam angles. Indeed, it dynamically explores angles and the only discretization that is necessary is due to the maximum accuracy that can be achieved by the linear accelerator machine. Experimental results are performed on phantom and real-life case studies, showing the advantages that come from our approach.

Motion management with phase-adapted 4D-optimization

Cancer treatment with ionizing radiation is often compromised by organ motion, in particular for lung cases. Motion uncertainties can significantly degrade an otherwise optimized treatment plan. We present a spatiotemporal optimization method, which takes into account all phases of breathing via the corresponding 4D-CTs and provides a 4D-optimal plan that can be delivered throughout all breathing phases. Monte Carlo dose calculations are employed to warrant for highest dosimetric accuracy, as pertinent to study motion effects in lung. We demonstrate the performance of this optimization method with clinical lung cancer cases and compare the outcomes to conventional gating techniques. We report significant improvements in target coverage and in healthy tissue sparing at a comparable computational expense. Furthermore, we show that the phase-adapted 4D-optimized plans are robust against irregular breathing, as opposed to gating. This technique has the potential to yield a higher delivery efficiency and a decisively shorter delivery time.

Nonconvex Robust Optimization for Problems with Constraints

We propose a new robust optimization method for problems with objective functions that may be computed via numerical simulations and incorporate constraints that need to be feasible under perturbations. The proposed method iteratively moves along descent directions for the robust problem with nonconvex constraints and terminates at a robust local minimum. We generalize the algorithm further to model parameter uncertainties. We demonstrate the practicability of the method in a test application on a nonconvex problem with a polynomial cost function as well as in a real-world application to the optimization problem of intensity-modulated radiation therapy for cancer treatment. The method significantly improves the robustness for both designs.

Universal scaling at field-induced magnetic phase transitions

We study field-induced magnetic order in cubic lattices of dimers with antiferromagnetic Heisenberg interactions. The thermal critical exponents at the quantum phase transition from a spin liquid to a magnetically ordered phase are determined from stochastic series expansion quantum Monte Carlo simulations. These exponents are independent of the interdimer coupling ratios, and converge to the value obtained by considering the transition as a Bose-Einstein condensation of magnons,

Robust optimization in electromagnetic scattering problems

{\ensuremath{\alpha}}_{\mathrm{BEC}}=\frac{3}{2}.

Robust chirped mirrors

The scaling results are of direct relevance to the spin-dimer systems

Bose-glass phases in disordered quantum magnets

{\mathrm{TlCuCl}}_{3}

Improving thin-film manufacturing yield with robust optimization

and