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Peer-reviewedCC BYFoundations of Computational Mathematics

First-Order Optimality Conditions for Non-Commutative Optimization Problems

  • Mateus Araújo1
  • Igor Klep2,3,4
  • Andrew J. P. Garner5
  • Tamás Vértesi6
  • Miguel Navascués5
6 affiliations
  1. 1Departamento de Física Teórica, Atómica y Óptica, Laboratory for Disruptive Interdisciplinary Science (LaDIS), Universidad de Valladolid, 47011, Valladolid, Spain
  2. 2Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia
  3. 3Famnit, University of Primorska, Koper, Slovenia
  4. 4Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
  5. 5Institute for Quantum Optics and Quantum Information (IQOQI) Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, Wien, 1090, Austria
  6. 6HUN-REN Institute for Nuclear Research, P.O. Box 51, H-4001, Debrecen, Hungary
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Abstract

We consider the problem of optimizing the state average of a polynomial of non-commuting variables, over all states and operators satisfying a number of polynomial constraints, and over all Hilbert spaces where such states and operators are defined. Such non-commutative polynomial optimization (NPO) problems are routinely solved through hierarchies of semidefinite programming (SDP) relaxations. By formulating the general NPO problem in Lagrangian terms, we heuristically derive first-order optimality conditions via small variations in the problem variables. Although the derivation is not rigorous, it gives rise to two types of optimality conditions—state and operator—which are rigorously analyzed in the paper. Both types of conditions can be enforced through additional positive semidefinite constraints in the SDP hierarchies. State optimality conditions are shown to be satisfied by all NPO problems. For NPO problems with optimal solutions (such as, e.g., Archimedean ones) they allow enforcing a new type of constraints: namely, restricting the optimization over states to the set of common ground states of an arbitrary number of operators. Operator optimality conditions are the non-commutative analogs of the Karush–Kuhn–Tucker (KKT) conditions, which are known to hold in many classical optimization problems. In this regard, we prove that a weak form of operator optimality holds for all NPO problems; stronger versions require the problem constraints to satisfy some qualification criterion, just like in the classical case (e.g.: Mangasarian–Fromovitz constraint qualification). We test the power of the new optimality conditions by computing local properties of ground states of many-body spin systems and the maximum quantum violation of Bell inequalities.

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Cite

BibTeXRIS
BibTeX
@article{araujo2026firstorder,
  title = {{First-Order Optimality Conditions for Non-Commutative Optimization Problems}},
  author = {Mateus Araújo and Igor Klep and Andrew J. P. Garner and Tamás Vértesi and Miguel Navascués},
  journal = {Foundations of Computational Mathematics},
  year = {2026},
  month = jul,
  doi = {10.1007/s10208-026-09761-x},
  url = {https://doi.org/10.1007/s10208-026-09761-x},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - First-Order Optimality Conditions for Non-Commutative Optimization Problems
AU  - Mateus Araújo
AU  - Igor Klep
AU  - Andrew J. P. Garner
AU  - Tamás Vértesi
AU  - Miguel Navascués
PY  - 2026
DA  - 2026/07/07
JO  - Foundations of Computational Mathematics
DO  - 10.1007/s10208-026-09761-x
UR  - https://doi.org/10.1007/s10208-026-09761-x
AB  - We consider the problem of optimizing the state average of a polynomial of non-commuting variables, over all states and operators satisfying a number of polynomial constraints, and over all Hilbert spaces where such states and operators are defined. Such non-commutative polynomial optimization (NPO) problems are routinely solved through hierarchies of semidefinite programming (SDP) relaxations. By formulating the general NPO problem in Lagrangian terms, we heuristically derive first-order optimality conditions via small variations in the problem variables. Although the derivation is not rigorous, it gives rise to two types of optimality conditions—state and operator—which are rigorously analyzed in the paper. Both types of conditions can be enforced through additional positive semidefinite constraints in the SDP hierarchies. State optimality conditions are shown to be satisfied by all NPO problems. For NPO problems with optimal solutions (such as, e.g., Archimedean ones) they allow enforcing a new type of constraints: namely, restricting the optimization over states to the set of common ground states of an arbitrary number of operators. Operator optimality conditions are the non-commutative analogs of the Karush–Kuhn–Tucker (KKT) conditions, which are known to hold in many classical optimization problems. In this regard, we prove that a weak form of operator optimality holds for all NPO problems; stronger versions require the problem constraints to satisfy some qualification criterion, just like in the classical case (e.g.: Mangasarian–Fromovitz constraint qualification). We test the power of the new optimality conditions by computing local properties of ground states of many-body spin systems and the maximum quantum violation of Bell inequalities.
SN  - 1615-3375
SN  - 1615-3383
ER  -