## 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 \(|H^{-1}r|\).