Expertise
& Research interests

*In
particular, I am interested in convex optimization –
minimizing or maximizing linear functionals over convex subsets
of (finite-dimensional real) vector spaces.*

*Linear
Programming (LP) –minimizing a linear functional over a set
of finitely many linear inequalities– is, perhaps, one of
the most well recognized problems of convex optimization today.
In 1975 L. Kantorovich received the Nobel prize in economics for
his work in linear programming duality theory; in 1980 and 1984,
the subject made the headlines and the front page of New York
Times, respectively, with a discovery of a polynomial-time
algorithm for LP by L. Khachian, and a discovery of a new
practically efficient polynomial-time interior-point method for
linear programming by N. Karmarkar. The existence of an algorithm
whose run-time is polynomial only in problem dimensions was cited
by S. Smale as one of the 18 greatest unsolved problems of the
21st century. *

*Many
numerical methods for convex optimization fall into category of
the so-called path-following methods; thus, a better
understanding of the underlying paths is essential to developing
more efficient optimization algorithms. For instance, any convex
optimization problem is, in theory, amenable to Interior-Point
Methods (IPM) – a family of particularly efficient
numerical algorithms for convex optimization. IPMs typically
follow the so-called central path that leads to an optimum.*

*My
thesis work has been primarily dedicated to studying
Shrink-Wrapping trajectories for a new class of optimization
methods for LP *[1]*,
and more generally, convex optimization problems posed over
hyperbolicity cones *[2]*.
More recently, I have pursued the study of the curvature of the
central path for Linear Programming *[3,
4]*.*

*In
five words, I consider medical applications of mathematics as
“math you feel good about”. It is an emerging field
where the main impact is yet to come. Applications to oncology
and, in particular, radiation therapy treatment design is an
important research area that I feel passionate about. The current
interdisciplinary research project I am involved in targets a
radical advance in the state-of-the-art application of modern
convex optimization methods for radiation therapy treatment
planning for cancer patients. Extremely high computational
demands associated with the optimal radiation therapy treatment
planning place this problem in the realm of novel applications
for which much more powerful optimization methods, models and
numerical software are desperately needed.*

*The
project pursues two main objectives:*

*a)
to minimize the adverse effect of the uncertainties in the
radiation therapy treatment planning phase via a combination of
image guidance and novel large-scale robust optimization
techniques *[5]*,*

*b)
to develop a suitable (convex optimization based) modeling
framework for optimal radiotherapy planning that incorporates
“dose-volume constraints”, amenable to
state-of-the-art computational tools *[6]*.*

*In
2007 I received Best Novel Use of Mathematics in Technology
Transfer MITACS
Award for research in optimal and robust radiotherapy design.*

*The
above mentioned Linear Programming may serve as a perfect example
of the intimate relationship between Mathematical Programming /
Operations Research and Computational and Discrete Geometry.
While LP is one of the most widely used decision making tools of
OR, the behavior of many numerical algorithms to solve such
problems is studied using the tools and methods of Computational
and Discrete Geometry, e.g., one studies the face-lattice of
polyhedra to understand the behavior of pivot-type methods.*

*Recently
I became interested in optimization-based bounds for Kissing
Numbers. Given an n-dimensional unit sphere centered at the
origin, the maximal number of unit spheres that can touch it from
the outside, with no mutual overlap, is called the Kissing Number
in dimension n. The Kissing Numbers are known only for dimensions
1, 2, 3, 4, 8 and 24. Determining the Kissing Number is a
classical problem in geometry and is of particular interest to
coding theory (spherical codes).*

*Typically
convex optimization problems do not admit closed-form analytic
solutions and numerical methods have to be used.*

*Nowadays,
one is capable of solving LP problems with millions of variables
on a decent computing station. However, for many novel and
existing applications there is a compelling need to solve larger
and more difficult convex, and not necessarily linear, problems.
For instance, the earlier mentioned optimal radiotherapy planning
falls within the realm of optimization problems whose
computational demands far-exceed the capacities provided by
state-of-the-art optimization software.*

*A
dominant computational cost per IPM iteration is attributed to
solving nearly dense system of linear equations. Since the
dimensions of the system correspond to the dimensions of the
underlying optimization problem, the need for high-performance
(distributed) linear algebra becomes imminent as the dimensions
increase.*

*We
target design and implementation of a modular platform suited for
development of high-performance IPM-based solvers. One of the
platform design goals is to provide a transparent access to
hardware-optimized Basic Linear Algebra Subroutines. For some
optimization problem classes we also investigate the use of
GP-GPUs.*

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