Research

Geometry that improves computation

I study how structural information in optimization models can be translated into stronger theory and more effective algorithms.

01

Facial reduction and singularity

Semidefinite programs often fail standard regularity conditions. Facial reduction identifies the smallest face containing the feasible region, exposing hidden structure and restoring a better-posed formulation. My work examines both algorithmic forms of facial reduction and the behavior of singularity degree.

Themes: exposing vectors, strict feasibility, maximum singularity degree, degeneracy, exact duality

02

Semidefinite relaxations

Semidefinite and doubly nonnegative relaxations provide strong bounds for hard discrete problems. I develop formulations that combine problem structure with reductions that improve numerical stability and tractability.

Applications: quadratic assignment, quadratic shortest path, graph partitioning, quadratically constrained quadratic programs

03

Large-scale computational methods

Strong relaxations are useful only if they can be solved at meaningful scale. I work on splitting methods, alternating direction methods, and structured projection techniques for large conic models.

Methods: ADMM, Peaceman–Rachford splitting, symmetry reduction, first-order optimization

The connecting idea

Across these directions, the central question is the same: how can geometry and combinatorial structure be used before and during optimization to produce a formulation that is both mathematically stronger and computationally better?