Correction to: On the Polyhedral Homotopy Method for Solving Generalized Nash Equilibrium Problems of Polynomials

Correction to: Journal of Scientific Computing (2023) 95:13. https://doi.org/10.1007/s10915-023-02138-0

The original version of the article unfortunately contained a mistake in three places caused by miscommunication in the process of publication. It has been corrected in this correction and the original article has been corrected.

• 1. On the 7th last line page 8 (Step 3 of Algorithm 3.1) in the original version of the article, it says:

  “If $V_i =\emptyset$ for all $i\in \{1, \dots , N\}$, or $f_i (v_i , u_{-i} )-\ldots$... for all $v_i\in V_i \cap X_i (u_{-i} )$, then go to the next step.Otherwise, go back to Step 2."

  This sentence is incorrect and should be corrected to

  “If $V_i =\emptyset$ for all $i\in \{1, \dots , N\}$, or $f_i (v_i , u_{-i} )-f_i (u_i , u_{-i})\ge 0$ for all $i\in \{1, \dots , N\}$ and for all $v_i\in V_i \cap X_i (u_{-i} )$, then go to the next step. Otherwise, go back to Step 2."

• 2. On the 4th last line on page 14, after “the variety”, there should be a “$\{$" before the $x_1\in\mathbb{C}^{n_1}$, and there should be a “$= 0$" after “$g^w_{1,m_1}(x_1,z_{-1})$." The sentence should be corrected to:

  "For a generic $z_{-1}\in\mathbb{C}^{\hat{n}-\hat{n}_1}$, [37, Proposition 2.2] implies that the variety $\{x_1\in\mathbb{C}^{\hat{n}_1}:g_{1,1}^w(x_1,z_{-1})=\dots=g_{1,m_1}^w(x_1,z_{-1})=0\}$ is smooth, i.e., the matrix $(\mbox{Jac}_i^w)^{\circ}$ has full column rank at $(x_1,z_{-1})$ for all $x_1\in\mathbb{C}^{\hat{n}_1}$."

• 3. For reference [22], the title should be corrected to:

  “$\texttt{GloptiPoly3}$: moments, optimization and semidefinite programming.".

• 4. In the beginning of Lemma 4.3, change “Let $p$ be a polynomial” to “Let $p$ be a dense polynomial”.