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This paper studies Positivstellensätze and moment problems for sets K that are given by universal quantifiers. Let Q be the closed set of universal quantifiers. Fix a finite nonnegative Borel measure whose support is Q and assume it satisfies the multivariate Carleman condition. First, we prove a Positivstellensatz with universal quantifiers: if a polynomial f is positive on K, then f belongs to the associated quadratic module, under the archimedeanness assumption. Second, we prove some necessary and sufficient conditions for a full (or truncated) multisequence to admit a representing measure supported in K. In particular, the classical flat extension theorem of Curto and Fialkow is generalized to truncated moment problems on such a set K. Third, we present applications of the above Positivstellensatz and moment problems in semi-infinite optimization, where feasible sets are given by infinitely many constraints with universal quantifiers. This results in a new hierarchy of Moment-SOS relaxations. Its convergence is shown under some usual assumptions. The quantifier set Q is allowed to be non-semialgebraic, which makes it possible to solve some optimization problems with non-semialgebraic constraints. Funding: X. Hu and J. Nie are partially supported by the NSF [Grant DMS-2110780]. I. Klep is supported by the Slovenian Research Agency program P1-0222 [also Grants J1-50002, J1-60011, J1-50001, J1-2453, N1-0217, and J1-3004] and was partially supported by the Marsden Fund Council of the Royal Society of New Zealand. I. Klep’s work was partly performed within the project COMPUTE, funded within the QuantERA II program that has received funding from the EU’s H2020 research and innovation program under the GA No 101017733.
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This content will become publicly available on March 14, 2026
Convergence of Augmented Lagrangian Methods for Composite Optimization Problems
Local convergence analysis of the augmented Lagrangian method (ALM) is established for a large class of composite optimization problems with nonunique Lagrange multipliers under a second-order sufficient condition. We present a new second-order variational property called the semistability of second subderivatives and demonstrate that it is widely satisfied for numerous classes of functions, which is important for applications in constrained and composite optimization problems. Using the latter condition and a certain second-order sufficient condition, we are able to establish Q-linear convergence of the primal-dual sequence for an inexact version of the ALM for composite programs. Funding: Research of the first author is partially supported by Singapore National Academy of Science via SASEAF Programme under the grant RIE2025 NRF International Partnership Funding Initiative. Research of the second author is partially supported by the National Science Foundation under the grant DMS 2108546.
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- Award ID(s):
- 2108546
- PAR ID:
- 10621521
- Publisher / Repository:
- Informs
- Date Published:
- Journal Name:
- Mathematics of Operations Research
- ISSN:
- 0364-765X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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