# A Quantum-Behaved Neurodynamic Approach for Nonconvex Optimization with Constraints
## 2 Problem Statement and Model Description  
Consider the following constrained optimization problem given by  
\[ \begin{split}\min f(x),\\ \text{subject to}& g_{i}(x)\leq 0,\ i=1,2,\cdots,m, \end{split}  \tag{1}\]  
where \(x\in\mathbb{R}^{n}\) is the decision vector, \(f:\mathbb{R}^{n}\rightarrow\mathbb{R}\) is a continuously differentiable function, \(g_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}\) (\(i=1,2,\cdots,m\)) are continuously differentiable functions, denoting the inequality constraints. \(f(x)\) and \(g_{i}(x)\) are not necessarily convex. The feasible region \(\Omega=\{x\in\mathbb{R}^{n}:g_{i}(x)\leq 0,i=1,2,\cdots,m\}\) is assumed to be a nonempty set. Denote \(g(x)=(g_{1}(x),g_{2}(x),\ldots,g_{m}(x))^{\top}\). Then, the problem (1) can be written as  
\[ \begin{split}\min f(x),\\ \text{subject to}& g(x)\leq 0.\end{split}  \tag{2}\]  
Let \(M_{1}\) be a lower bound of the optimal value of \(f(x^{*})\) in the problem (1), i.e., \(M_{1}\leq f(x^{*})\), where \(x^{*}\) is an optimal solution of (1). Denote \(d(x,M_{1})=f(x)-M_{1}\) and \(F(x,M_{1})=0.5d(x,M_{1})(d(x,M_{1})\ +\ |d(x,M_{1})|)\). Consider an energy function  
\[E(x)=F(x,M_{1})+\gamma\big{(}g(x)^{\top}\text{sgn}(g(x)+|g(x)|)\big{)}, \tag{3}\]  
where \(\gamma>0\) is a penalty parameter, and \(\text{sgn}(\cdot)\) is the sign function. Then, inspired by [14; 22], we construct the following feedback neural network for minimizing \(E(x)\)  
\[ \begin{split}\dot{x}=&-\
abla E(x)\\ =&-\
abla F(x,M_{1})-\gamma\big{(}\
abla g(x)^{ \top}\text{sgn}(g(x)+|g(x)|)\big{)}.\end{split}  \tag{4}\]  
**Remark 1**. : _In [14], [PERSON] et al. proposed the energy function \(G(x,M_{1})=F(x,M_{1})+\frac{1}{2}g(x)^{\top}(g(x)-|g(x)|)+\frac{1}{2}\ \|\ Ax-b\ \|^{2}\) for the convex nonlinear programming problem_  
\[ \begin{array}{ll}\min&f(x),\\ \text{subject to}&g(x)\geq 0,\\ &Ax=b.\end{array} \]  
_In fact, for some equality constraint \(h(x)=0\), one can transform the equality constraint into inequality constraints \(-h(x)\leq 0\) and \(h(x)\leq 0\). The energy function \(E(x)\) in (3) contains a penalty parameter \(\gamma\). Such a penalty \(\gamma\) can strengthen the constraints and make the results converge into the constraints more efficiently in optimization problems with inequality constraints. In addition, as we will show later, the penalty parameter \(\gamma\) makes the neural network feasible to fit in the network model for solving some nonconvex problems in [22]._To derive the convergence results of the feedback neural network (4), we first introduce the following definitions. **Definition 1**. : _[_22_]_ _Suppose that \(f\) is a differentiable function and defined on an open convex set \(M\subset\mathbb{R}^{n}\). Then, \(f\) is quasiconvex if \(a,b\in M,f(a)\geq f(b)\Rightarrow\
abla f(a)^{\top}(b-a)\leq 0\)._  
**Definition 2**. : _[_22_]_ _Suppose that \(f\) is a differentiable function and defined on an open convex set \(M\subset\mathbb{R}^{n}\). Then, \(f\) is pseudoconvex if \(a,b\in M,f(a)>f(b)\Rightarrow\
abla f(a)^{\top}(b-a)<0\)._  
**Theorem 1**. : _If \(f(x)\) in (2) is a pseudoconvex function, then \(F(x,M_{1})\) in (3) is also a pseudoconvex function._  
Proof of Theorem 1.: Let us rewrite \(F(x,M_{1})=0.5d(x,M_{1})(d(x,M_{1})+|d(x,M_{1})|)\) as  
\[F(x,M_{1})=\left\{ \begin{array}{ll}d(x,M_{1})^{2},&d(x,M_{1})\geq 0,\\ 0,&d(x,M_{1})<0.\end{array} \right.\]  
By definition of \(d(x,M_{1})\), i.e., \(d(x,M_{1})=f(x)-M_{1}\), we have  
\[\
abla F(x,M_{1})=\left\{ \begin{array}{ll}2d(x,M_{1})\
abla f(x),&d(x,M_{1}) \geq 0,\\ 0,&d(x,M_{1})<0,\end{array} \right. \tag{5}\]  
and  
\[F(x_{1},M_{1})-F(x_{2},M_{1})=\left\{ \begin{array}{ll}d(x_{1},M_{1})^{2}-d(x _{2},M_{1})^{2},&d(x_{1},M_{1})\geq 0,d(x_{2},M_{1})\geq 0,\\ d(x_{1},M_{1})^{2},&d(x_{1},M_{1})\geq 0,d(x_{2},M_{1})<0,\\ -d(x_{2},M_{1})^{2},&d(x_{1},M_{1})<0,d(x_{2},M_{1})\geq 0,\\ 0,&d(x_{1},M_{1})<0,d(x_{2},M_{1})<0.\end{array} \right.\]  
Note that \(F(x_{1},M_{1})-F(x_{2},M_{1})>0\) holds when  
\[F(x_{1},M_{1})-F(x_{2},M_{1})=\left\{