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.
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