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This paper points out the error in the HJB equation in a previous published paper and presents the correct HJB equation. We show that the Hamilton-Jacobi-Bellman (HJB) equation for the salesperson's problem is not correct because of the effect of a positive discount factor for the infinite-horizon risk-sensitive control. As a result, their analyses and insights are incorrect. We present the correct HJB equation for the salesperson's problem, which is a partial differential equation. Thus, further investigation on the correct solutions is needed to better understand the problem. History: K. Sudhir served as the senior editor for this article. Funding: X. Kong acknowledges financial support from the Natural Science Foundation of China [Grants 72101243 and 72091212/72091210] and the Fundamental Research Funds for the Central Universities [Grant WK2040000033]. The research of Y. Yu is partially supported by the Hong Kong Research Grants General Proposal Fund [Grant 9043432], CityU [Grants 7005793 and 7005591], and the Laboratory for AI-Powered Financial Technologies. [ABSTRACT FROM AUTHOR]

In this paper, we consider optimization problems over closed embedded submanifolds of Rn , which are defined by the constraints c(x) = 0. We propose a class of constraint-dissolving approaches for these Riemannian optimization problems. In these proposed approaches, solving a Riemannian optimization problem is transferred into the unconstrained minimization of a constraint-dissolving function (CDF). Different from existing exact penalty functions, the exact gradient and Hessian of CDF are easy to compute. We study the theoretical properties of CDF and prove that the original problem and CDF have the same first-order and second-order stationary points, local minimizers, and Łojasiewicz exponents in a neighborhood of the feasible region. Remarkably, the convergence properties of our proposed constraint-dissolving approaches can be directly inherited from the existing rich results in unconstrained optimization. Therefore, the proposed constraint-dissolving approaches build up short cuts from unconstrained optimization to Riemannian optimization. Several illustrative examples further demonstrate the potential of our proposed constraint-dissolving approaches. Funding: The research of N. Xiao and K.-C. Toh is supported by the Ministry of Education of Singapore Academic Research Fund Tier 3 [Grant MOE-2019-T3-1-010]. The research of X. Liu is supported in part by the National Natural Science Foundation of China [Grants 12125108, 11971466, 12288201, 12021001, and 11991021]; the National Key R&D Program of China [Grants 2020YFA0711900 and 2020YFA0711904]; and the Key Research Program of Frontier Sciences, Chinese Academy of Sciences [Grant ZDBS-LY-7022]. [ABSTRACT FROM AUTHOR]