Characterisation of zero duality gap for optimization problems in spaces without linear structure
We prove sufficient and necessary conditions ensuring zero Lagrangian duality gap for Lagrangians defined with help of general perturbation functions. This kind of Lagrangians include generalized and augmented Lagrangians. To this aim, we use the Φ -convexity theory and we formulate our zero duality...
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| Published in | Journal of global optimization Vol. 92; no. 1; pp. 135 - 158 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.05.2025
Springer Nature B.V |
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| Abstract | We prove sufficient and necessary conditions ensuring zero Lagrangian duality gap for Lagrangians defined with help of general perturbation functions. This kind of Lagrangians include generalized and augmented Lagrangians. To this aim, we use the
Φ
-convexity theory and we formulate our zero duality gap conditions in terms of elementary functions
φ
∈
Φ
. The obtained results apply to optimization problems involving prox-bounded functions, DC functions, weakly convex functions and paraconvex functions as well as infinite-dimensional linear optimization problems, including Kantorovich duality which plays an important role in determining Wasserstein distance. |
|---|---|
| AbstractList | We prove sufficient and necessary conditions ensuring zero Lagrangian duality gap for Lagrangians defined with help of general perturbation functions. This kind of Lagrangians include generalized and augmented Lagrangians. To this aim, we use the Φ-convexity theory and we formulate our zero duality gap conditions in terms of elementary functions φ∈Φ. The obtained results apply to optimization problems involving prox-bounded functions, DC functions, weakly convex functions and paraconvex functions as well as infinite-dimensional linear optimization problems, including Kantorovich duality which plays an important role in determining Wasserstein distance. We prove sufficient and necessary conditions ensuring zero Lagrangian duality gap for Lagrangians defined with help of general perturbation functions. This kind of Lagrangians include generalized and augmented Lagrangians. To this aim, we use the Φ -convexity theory and we formulate our zero duality gap conditions in terms of elementary functions φ ∈ Φ . The obtained results apply to optimization problems involving prox-bounded functions, DC functions, weakly convex functions and paraconvex functions as well as infinite-dimensional linear optimization problems, including Kantorovich duality which plays an important role in determining Wasserstein distance. |
| Author | Syga, Monika Bednarczuk, Ewa |
| Author_xml | – sequence: 1 givenname: Ewa surname: Bednarczuk fullname: Bednarczuk, Ewa organization: Systems Research Institute, Polish Academy of Sciences, Faculty of Mathematics and Information Science, Warsaw University of Technology – sequence: 2 givenname: Monika orcidid: 0000-0001-8816-7558 surname: Syga fullname: Syga, Monika email: monika.syga@pw.edu.pl organization: Faculty of Mathematics and Information Science, Warsaw University of Technology |
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| Keywords | Abstract convexity Lagrangian duality Prox-regular functions Minimax theorem Weak duality Augmented Lagrangians Wasserstein distance Paraconvex and weakly convex functions Kantorovich duality Zero duality gap |
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| SubjectTerms | Computer Science Convex analysis Convexity Lagrangian function Mathematical functions Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Real Functions |
| Title | Characterisation of zero duality gap for optimization problems in spaces without linear structure |
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