# Methods for Constrained Optimization Problems with Box Constraints

Consider this problem:

\begin{equation} \begin{array}{ll} \text{minimize } & f(x) \\ \text{subject to } & a \leq x \leq b \end{array} \end{equation}

where $a,b,x \in \mathbb{R}^n$ and $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is continuously differentiable. By this definition, the feasible set is closed and convex.

I would greatly appreciate it if you could give me a big picture of the classes of methods that solve this problem. I know there are some methods such as projected gradient/Newton methods or primal-dual active set/semismooth Newton methods and some variations of these methods. However, these are specific. The book by Nocedal and Wright seems to discuss the following types of methods:

• active set
• interior-point
• penalty and augmented Lagrangian

The book "Optimization with PDE Constraints" by Hinze, Pinnau, and Ulbrich discusses:

• Newton's methods
• Semismooth Newton's methods (which Google Scholar says is equivalent to the primal-dual active set methods)

I am doing a literature review of the methods for this problem and hope that through my literature review I can discover something that needs improving. I have only 2 months (January and February) to do this. I am very new to this field and do not consider myself mathematically mature. I have very little experiences. Hence, it would be great if some of you could kindly share your experiences/visions. It would help me a lot if I could have a big picture of the methods. I would forever be grateful.

• Your definition actually describes an open (not closed*) domain. You'd need $a\le x\le b$. – Wolfgang Bangerth Jan 3 '15 at 1:56
• yes. you were right. typo :D – Linh Huynh Jan 3 '15 at 2:01