From Continuous to Discrete

Abstract

One approach that has been used to solve the DGP is to represent it as a continuous optimization problem [59]. To understand it, we consider a DGP with K = 2, V = {u, v, s}, E = {{ u, v}, {v, s}}, where the associated quadratic system is

$$\displaystyle\begin{array}{rcl} (x_{u1} - x_{v1})^{2} + (x_{ u2} - x_{v2})^{2}& =& d_{ uv}^{2} {}\\ (x_{v1} - x_{s1})^{2} + (x_{ v2} - x_{s2})^{2}& =& d_{ vs}^{2}, {}\\ \end{array}$$

which can be rewritten as

$$\displaystyle\begin{array}{rcl} (x_{u1} - x_{v1})^{2} + (x_{ u2} - x_{v2})^{2} - d_{ uv}^{2}& =& 0 {}\\ (x_{v1} - x_{s1})^{2} + (x_{ v2} - x_{s2})^{2} - d_{ vs}^{2}& =& 0. {}\\ \end{array}$$

Consider the function $f: \mathbb{R}^{6} \rightarrow \mathbb{R}$, defined by

$$\displaystyle\begin{array}{rcl} f(x_{u1},x_{u2},x_{v1},x_{v2},x_{s1},x_{s2})& =& \left ((x_{u1} - x_{v1})^{2} + (x_{ u2} - x_{v2})^{2} - d_{ uv}^{2}\right )^{2} {}\\ & +& \left ((x_{v1} - x_{s1})^{2} + (x_{ v2} - x_{s2})^{2} - d_{ vs}^{2}\right )^{2}. {}\\ \end{array}$$

It is not hard to realize that the solution $x^{{\ast}}\in \mathbb{R}^{6}$ of the associated DGP can be found by solving the following problem:

$$\displaystyle{ \min _{x\in \mathbb{R}^{6}}f(x). }$$

(3.1)

That is, we wish to find the point $x^{{\ast}}\in \mathbb{R}^{6}$ which attains the smallest value of f.