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.
List of publications, notes, thoughts, etc.