## Publications

[1] A Globally Convergent Primal-Dual Active-Set Framework for Large-
Scale Convex Quadratic Optimization

F. E. Curtis, **Z. Han**, and D. P. Robinson, * Computational Optimization and Applications *(COAP), 2014.

[2] Globally Convergent Primal-Dual Active-Set Methods with Inexact Subproblem Solves

F. E. Curtis, **Z. Han**, * Optimization Online*, 2014(under review).

## Notes

On Optimization[1] A Note on Inexact Newton's Method

- A proof of global convergence in solving unconstrained nonlinear optimization problem \(\min_{x\in\mathbb R^n}\ f(x)\) by applying inexact Newton's method on \(\nabla f(x) = 0\).

[2] Counterexamples in plain primal-dual active-set method

- For mathematicians, the greatest despair might be proposing a deceptively simple conjecture yet has to disprove it after many months of travail. But coming up empty-handed is a vital and oft-overlooked part of research. Each negative result rules out certain theories and strengthens others, shrinking the conceptual space in which the reality can be hiding.

On Numerical Methods

[1] Bounds on the Inverse of Nonnegative Matrix

- An interactive play to show that \(f_n:=\displaystyle\sum_{i=0}^n (I-H)^n\) approximates \(H^{-1}\in\mathbb R^{n\times n}\) in the limit. Certain assumptions on \(H\) yields the interesting interlacing behavior that \(\cdots < f_{2k+1} < f_{2k+3} <\cdots < H^{-1} < \cdots < f_{2k+2} < f_{2k}<\cdots\) for \(k\in \mathbb N\).

[2] Submatrix's Inverse and Inverse's Submatrix: A Note on M-matrix

- Suppose \(H\in\mathbb R^{n\times n}\succ 0\). Can we decide which one of \( \|(H^{-1})_{\mathcal I\mathcal I}\| \) and \( \|(H_{\mathcal I\mathcal I})^{-1} \| \) is larger? Yes, if \(H\) is an \(M\)-matrix or \(\|\cdot\|\) is Euclidean norm; No, at least for some positive definite matrices on \(\|\cdot \|_{1}\) or \(\|\cdot \|_{\infty}\).

[3] An application of Hölder’s inequality

- Hölder’s inequality is applied to obtain a cheaply computable upper bound of \(|H^{-1}r|\).

## Slides

[1] A Primal-Dual Active-Set Algorithm for a Class of Large-Scale Convex Quadratic Optimization Problems

To appear for

*INFORMS Annual Conference*. San Francisco, CA. Nov 2014.