# A Quantum-Behaved Neurodynamic Approach for Nonconvex Optimization with Constraints
## 2 Problem Statement and Model Description
\begin{array}{ll}d(x_{1},M_{1})^{2}-d( x_{2},M_{1})^{2}>0,&d(x_{1},M_{1})\geq 0,d(x_{2},M_{1})\geq 0,\\ d(x_{1},M_{1})^{2}>0,&d(x_{1},M_{1})\geq 0,d(x_{2},M_{1})<0.\end{array} \right.\]  
Using the definition of \(d(x,M_{1})\) again, it follows that \(F(x_{1},M_{1})-F(x_{2},M_{1})>0\) implies that both \(d(x_{1},M_{1})>0\) and \(f(x_{1})>f(x_{2})\) holds. Therefore, using the fact that \(f(x)\) is a pseudoconvex function, \(F(x_{1},M_{1})-F(x_{2},M_{1})>0\) also implies \(\
abla f(x_{1})(x_{2}-x_{1})<0\). In addition, since \(F(x_{1},M_{1})\ -\ F(x_{2},M_{1})>0\) implies \(d(x_{1},M_{1})>0\), it follows from (5) that  
\[\
abla F(x_{1},M_{1})^{\top}(x_{2}-x_{1})=2d(x_{1},M_{1})\
abla f(x_{1})(x_{2 }-x_{1})<0\]  
if \(F(x_{1},M_{1})-F(x_{2},M_{1})>0\). The proof is completed. **Remark 2**. : _From the proof of Theorem 1, it should be noted that the function \(F\) is constructed as a semi-positive definition function and the term \(|d(x,M_{1})|\) can be seen as an adaptive adjustment function when computing the gradient of \(E(x)\). That is, when \(f(x)\) approaches \(f(x^{*})\), \(d(x,M_{1})\) becomes smaller and \(\
abla f(x)\) plays a minor role in computing the gradient of \(E(x)\); when \(f(x)\) is far away from \(f(x^{*})\), \(d(x,M_{1})\) becomes larger and \(\
abla f(x)\) plays an important role in computing the gradient of \(E(x)\)._  
To establish a relationship between the feedback neural network (4) and the neural network in [22], we consider the following notion and lemmas. **Definition 3**. : _[_29_]_ _([PERSON]'s generalized gradient) The generalized gradient of \(f\) at \(x\), denoted by \(\partial f(x)\), is the convex hull of the set of limits of the form \(\lim\
abla f(x+l_{i})\), where \(l_{i}\to 0\) as \(i\to\infty\)._  
**Lemma 1**. : _[_22_]_ _[Proposition 6]_ _Let \(g(x)\) be continuously differentiable. Then, \(\max\left\{0,g(x)\right\}\) is a regular function, its [PERSON]'s generalized gradient is_  
\[\partial\max\left\{0,g(x)\right\}=\left\{ \begin{array}{ll}\
abla g(x),&g(x)>0,\\ a_{g}\
abla g(x),&g(x)=0,\\ 0,&g(x)<0,\end{array} \right.\]  
_where \(\alpha_{g}\in[0,1]\)._  
**Lemma 2**. : _[_22_]_ _[Theorem 4]_ _For the problem (1), if one of the following two conditions holds,_  
1. \(f(x)\) _and_ \(g_{i}(x)\)_,_ \(i=1,2,\cdots,m\) _are convex functions;_
2. \(f(x)\) _is a pseudoconvex function and_ \(g_{i}(x)\)_,_ \(i=1,2,\cdots,m\) _are quasiconvex functions,_  
_then, for a sufficiently small penalty factor \(\sigma>0\), any state of the following recurrent neural network_  
\[\dot{x}\in-\
abla f(x)-\tfrac{1}{\sigma}\partial\sum\limits_{i=1}^{m}\max \left\{0,g_{i}(x)\right\} \tag{6}\]  
_converges to an optimal solution of problem (1)._  
**Definition 4**. : _(KKT Point) For the problem (1), suppose that \(f(x):\mathbb{R}^{n}\to\mathbb{R}\) is differentiable and \(g_{i}(x):\mathbb{R}^{n}\to\mathbb{R},i=1,2,\cdots,m\) are differentiable functions. If \((\xi,\mu)\in\mathbb{R}^{n}\times\mathbb{R}^{m}\) satisfies the following conditions:_  
\[\
abla f(\bar{x})+\bar{\mu}^{\top}\
abla g(\bar{x})=0,\bar{\mu}\geq 0,g(\bar{x}) \leq 0,\bar{\mu}^{\top}g(\bar{x})=0,\]  
_then \(\bar{x}\) is said to be a KKT point of problem (1). The KKT conditions provide first-order necessary conditions for nonlinear programming problem and can be considered as local optimization points for programming problem._  
Under some acceptable assumptions (see [22] for details), the following properties hold. **Proposition 1**. : _[_22_]_ _[Theorem 1]_ _When the penalty parameter \(\sigma\) is sufficiently small, then any state of (6) is guaranteed to be convergent to the feasible region in finite time and stay there thereafter._  
**Proposition 2**.