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This paper deals with the geometric numerical integration of gradient flowand its application to optimization. Gradient flows often appear as modelequations of various physical phenomena, and their dissipation laws areessential. Therefore, dissipative numerical methods, which are numericalmethods replicating the dissipation law, have been studied in the literature.Recently, Cheng, Liu, and Shen proposed a novel dissipative method, theLagrange multiplier approach, for gradient flows, which is computationallycheaper than existing dissipative methods. Although their efficacy isnumerically confirmed in existing studies, the existence results of theLagrange multiplier approach are not known in the literature. In this paper, weestablish some existence results. We prove the existence of the solution undera relatively mild assumption. In addition, by restricting ourselves to aspecial case, we show some existence and uniqueness results with concretebounds. As gradient flows also appear in optimization, we further apply thelatter results to optimization problems.

Existence results on Lagrange multiplier approach for gradient flows and application to optimization