## Optimization

A Globally Convergent Primal-Dual Active-Set Framework for Large- Scale Convex Quadratic Optimization
- Authors: F. E. Curtis, Z. Han, and D. P. Robinson, COR@L Laboratory, Department of ISE, Lehigh University, 12T-013, 2012. Proposed a enhanced primal-dual active-set algorithm with the ability to solve general convex QPs of the form $\min_{x\in\mathbb R^n} \{ \frac{1}{2}x^THx + c^Tx:\; Ax = b,\; \ell\leq x\leq u\}.$
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 $$\nabla f(x) = 0$$.
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.

## Algebra, Matrix Theory

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 $$k\in \mathbb N$$.

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 $$\|\cdot \|_{\infty}$$.

An application of Hölder’s inequality
- Hölder’s inequality is applied to obtain a cheaply computable upper bound of $$|H^{-1}r|$$.

List of publications, notes, thoughts, etc.