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LIT-1212: Multiple-gradient descent algorithm (MGDA) for multiobjective optimization

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StatusActive — read 2026-10-29 (NOTE-1401) in full, from the Comptes Rendus Mathématique's own open copy of the 2012 note; worth reading as the short published source for Pareto-stationarity and for the minimum-norm point of the gradients' convex hull as a common descent direction, the constructive half of CLAIM-366's convex-hull test. Read it knowing that it does not state the converse (zero in the hull excludes common strict descent), that its convergence theorem's proof is a sketch, and that the direction, unlike its existence, depends on how the gradients are scaled, which the author leaves open.
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Tagsmathematics, operations-research, anthology-candidate
Cited as “paper” byNOTE-1401 — Désidéri — Multiple-gradient descent algorithm (MGDA)

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Désidéri, J.-A. (2012). Multiple-gradient descent algorithm (MGDA) for multiobjective optimization. Comptes Rendus Mathématique 350(5–6): 313–318. doi:10.1016/j.crma.2012.03.014. First as INRIA Research Report RR-6953, “Multiple-Gradient Descent Algorithm (MGDA)”, June 2009.

Read in full on 2026-10-29 (NOTE-1401).

Why this is here

A33 of the exchange used its criterion, without a source: “If zero lies in the convex hull of the task gradients, there is no common strict first-order descent direction” (0 ∉ conv{g_1,…,g_n}). A34’s dual certificate and A35’s residual build on it. Bears on CLAIM-366 (common first-order descent is invariant under reparameterization; the sign of a gradient inner product is not) and CLAIM-365 (the residual’s dual certificate and governance regret).

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