On Debreu-Koopmans Theorem for Additively Decomposed Quasiconvex Functions and Its Applications

Time: 09h00-09h45, Tuesday 18.08.2026

Location: Room 508, building A6, Institute of Mathematics (18 Hoang Quoc Viet, Nghia Do, Ha Noi).

Abstract: The Debreu-Koopmans theorem [1] shows that if a separable function $h(x_1, \ldots, x_n) = \sum^n_{i=1} h_i (x_i)$ is quasiconvex, then at most one $h_i$ might be nonconvex and, as a consequence, when all components $h_i$ are quasiconvex, $h$ is not quasiconvex and it is not known to which class of functions $h$ belongs, giving rise to the Debreu-Koopmans problem. In this talk, we introduce the class of (strongly) star quasiconvex functions [4] and we prove that $h$ is (strongly) star quasiconvex if and only if all components $h_i$ are (strongly) star quasiconvex [2] and, as a consequence, we solves the Debreu-Koopmans problem. This result has strong implications in economic theory, since formally bridges classical diversification theory with behavioral economics, as it implies that separable $S$-shaped value (Sigmoid) functions from Prospect Theory [3] are star quasiconvex.  Finally, and if the time allows us, we present potential applications in economics and block-coordinate optimization.

1. G. Debreu, T.C. Koopmans. Additively decomposed quasiconvex functions. Math. Programm. 24:1-38, 1982.
2. F. Lara. On Debreu-Koopmans theorem for additively decomposed quasiconvex functions. arXiv: 2603.17088, 2026.
3. D. Kahneman, A. Tversky. Prospect theory: an analysis of decision under risk. Econometrica. 47(2):263-292, 1979.
4. P.Q. Khanh, F. Lara. Star quasiconvexity: a unified approach for linear convergence of first-order methods beyond convexity. arXiv: 2510.24981, 2025.