On the solution of systems of quadratic equations modeling load distribution in electric circuits (Q1902854)

Consider a system of algebraic equations \[ \begin{aligned} X(A_x x+ A_y y+ b_p)- Y(A_y x- A_y+ b_m) & = R_x,\tag{1}\\ X(A_y x- A_x y+ b_m)+ Y(A_x x+ A_y y+ b_p) & = R_y,\end{aligned} \] where \(A_x= (A_{xij})\), \(A_y= (A_{yij})\) are square \(n\)-dimensional matrices, \(b_m\), \(b_p\), \(R_x\), \(R_y\) are \(n\)-dimensional vectors \(X= \text{diag}(x)\), \(Y= \text{diag}(y)\), \(z= (x^T, y^T)^T\) are variables. All matrices, vectors and variables are real-valued. The paper deals with two reductions of computations when solving equation (1) to a concave programming problem (CPP). First, equation (1) is rewritten into the identity \(ab= [(a+ b)^2- (a- b)^2]/4\) for \(a\), \(b\) being real numbers, and further introducing two artificial variables. Necessary conditions of compatibility between the solutions of equation (1) and this CPP are given in terms of its parameters. The solution proceeds for fixed parameters of the CPP as the following problem: \[ \max \varphi_s(z)= \sum_k \varphi_k(z)= z^T Q_s z+ q^T_s z- r_s\tag{2} \] s.t. \(z\in D= \{z: \varphi_k(z)\leq 0, k= 1,\dots, 2n\}\), where \(\varphi_k(z)= z^T Q(w^k)z+ q^T(w^k)z- r(w^k)\), \(Q(w^k)= \sum^{2n}_{i= 1} w_i kQ^i\), \(z^T Q^iz= x_i F_{pi}- y_i F_{mi}\) and \(z^T Q^{i+ n}z= x_i F_{pi}+ y_i F_{pi}\) are given by \(F_{pi}= \sum_j A_{xij} x_j+ A_{yij} y_j\) and \(F_{mi}= \sum_j A_{yij} x_j- A_{xij} y_j\) for \(i= 1,\dots, n\). Deleting index \(k\) for simplicity, vectors \(q= [(Ub_p+ Vb_m)^T\), \((- Ub_m+ Vb_p)^T]^T\) and \(r= z^T[Q(w) z+ q]^T\), where \(w=(u^T,v^T)^T\), \(U= \text{diag}(u)\), \(V= \text{diag}(v)\), \(W= \text{diag}(w)\). To localize the solution of equation (1), a construction of sequences of \(w^k\) for equation (2) is given within sets of suitable combinations \(W^+ =\{w^k: w^k\in E^{2n}, \lambda_m(x)> 0\}\), \(\lambda_m(\cdot)\) denotes the minimal eigenvalue of \((\cdot)\). Procedures for cutting-off local solutions of equation (2) are given. Equation (1) is a model of the power balance problem in power networks.