# SOAL Research

**SOAL**focuses on the development of optimization methods for scientific computing, broadly construed. In addition to the theoretical work on the core disciplines of mathematical optimization, the resulting algorithms and techniques are implemented in the

*SOAL optimizer*and applied to a broad range of interdisciplinary problems spanning several application areas.

## Computational and Systems Biology

Computational and systems biology is the field of science that attempts to understand biological systems by applying data-analytical and computational tools and techniques. One of the current approaches to model biological systems as mathematical objects is mainly based on formulating biological information as mathematical constraints, thus called constraint-based modelling. In this way, many biological processes may directly be translated to optimization problems where the living organism is assumed to be trying to do some task as efficiently as possible without violating these constraints.

From another viewpoint, some of the research problems we work on in this lab are closely related to finding the most consistent model with respect to experimental data and given constraints that also require optimization techniques to be solved. In both regards, designing novel methods and algorithms for optimization, specifically in networks and network-related biological data, is of special interest in this lab. For instance, modelling metabolic networks by constraint-based reconstruction and analysis (COBRA) is one of the areas of interest in this line of research.

### Related courses

### Current projects

Context-Specific Reconstruction and Gap-Filling of Metabolic Networks by Sparse Reconciliation of Data Inconsistencies (Ali Fathi)

Designing a Database for Genome-Scale Metabolic Networks and its Applications in High-Throughput Data Analysis (Iman Ghadimi)

Exploring Pivot Genes and Clinical Prognosis using Combined Bioinformatics Approaches in the Colon Cancer (Ayoub Vaziri)

Geometric Optimization Techniques in Phylogeny Inference Problem (Hamed Vasei)

Machine and Deep Learning meet Genome-Scale Metabolic Modeling (Nahid Sarem)

Metabolic Engineering for Strain Design (Soraya Mirzaei)

Multi-Agent Based Simulation Framework (Fateme Alimirzaei)

### Useful resources

## Geometric and Variational Optimization

*Optimization* is distinguished to be among the most fundamental topics of science and technology, since, at least, rules of nature are usually, and astonishingly, based on *optimal* configurations or situations. It is also quite interesting that, almost accidentally and historically, scientists has set a priority on the concept of *approximation* rather than other features of human science related to computation, that over the past three centuries has already given rise to a well developed calculus of manifolds as the most general *continuous* setup in which one may study well-defined models of natural phenomena with an ability to do approximations, almost all of which are based on optimization.

As one of our latest view-points in science, one may simply mention that the principle of *optimized action* is the most fundamental cornerstone of our modern understanding of nature although scientists are still looking for a better replacement that may unify the whole *quantum* and *gravity* theories under the same umbrella. From this angle, most of what have already been developed in geometry of manifolds which are related to real-world models may be regarded as very general optimization problems in a *continuous* setup designed to facilitate *approximation*.

Most recently, and in juxtaposition to the older *continuous* framework of smooth manifolds, it has turned out that some *discrete* geometric models are also most natural to formalize real-world problems, and in some sense even fit better to the actual reality, while there has also been very strong motivations in mathematics itself to study non-smooth geometric objects in a very broad sense. In particular, discrete models based on *graphs*, as metric measure spaces, have set forth an arena of research motivated by actual models of natural phenomena as in biology, modern physics and chemistry. On the other hand, it is quite a surprise that such discrete geometric models also provide the most natural setup to analyze datasets, in what is usually referred to as *data science* as one of the most important scientific topics of our time.

All these strong and fundamental motivations along with a lot more appearing in a variety of theoretical and applied sciences (e.g. number theory, geometry, information theory, communication theory and coding, control theory, nanotechnology, theory of partition functions and statistical physics, etc.) have fixed *geometric optimization* as one of the most important topics of our scientific era.

It also ought to be mentioned that such optimization problems, although already having a natural geometric flavour, are quite hard to handle and are usually quite intractable. A very general, and historically old and effective, approach to tackle such problems is through designing some simple dynamical systems tightly related to the intrinsic geometry of the object and converging to the optimum solution or an approximation of it in time. It also turns out that such *iterative* or *variational* methods are not only quite important in practice but they also serve as very strong methods of proof in one's analysis of such problems. This whole scenario, in a way, has already formed one of the most fascinating parts of our current mathematical sciences with very strong ties to the *theory of large deviations, information theory, modern physics, graph theory* and *theory of metric measure spaces, partial differential equations, systems biology, learning theory* and *computer science*.

In particular, appearance of *partition functions* in a variety of diverse contexts as *Tutte polynomials* in graph theory, *machine learning*, *statistical physics*, *complex systems*, *knot theory*, *information theory*, *graphical models* and *filtering*, etc. and the importance of computing the *optimum values* in these settings, has already served as a very strong motivation for the development of *variational methods* and their deep applications in a number of most important practical topics in artificial intelligence and engineering.

The multifaceted theoretical nature of this field of study along with its diverse and effective applications in modern science and technology has turned this subject to an actual laboratory for scientific/algorithmic experimentation based on natural observations in which very interesting conjectures may crop up to motivate some later theoretical advances along with more profound applications.

### Related courses

### PhD students

Morteza Alimi

Amir Azizi

Mohammadhossein Shojaedin

### Some useful links

Handbook of Variational Methods for Nonlinear Geometric Data

Variational Methods for Machine Learning with Applications to Deep Networks

Graphical Models, Exponential Families, and Variational Inference

**?SOAL**. Last modified: April 07, 2022. Website built with Franklin.jl and the Julia programming language.