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Una aproximación segura semidefinida positiva de restricciones distribucionalmente robustas multivariadas

Jana Dienstbier1, Frauke Liers1, Jan Rolfes2,1

  • 1Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany.

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Resumen
Este resumen es generado por máquina.

Este estudio introduce una aproximación segura para problemas de optimización de distribución intratable y robusta (DRO) con funciones no convexas. El método permite el cálculo de soluciones probadamente robustas para modelos complejos de DRO.

Palabras clave:
Optimización distributivamente robustaOptimización de enteros mixtosOptimización robustaOptimización estocástica

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Área de la Ciencia:

  • Optimización
  • Investigación de las operaciones
  • Aprendizaje automático

Sus antecedentes:

  • Los problemas de optimización distribucionalmente robusta (DRO) con no convexidades a menudo son computacionalmente intratables debido a las restricciones duales semi-infinitas.
  • Los métodos existentes generalmente requieren suposiciones fuertes como la convexidad o la concavidad, lo que limita su aplicabilidad.
  • Los trabajos anteriores abordaron funciones univariadas, dejando en gran medida inexplorados los casos multivariados.

Objetivo del estudio:

  • Desarrollar una aproximación segura para las reformulaciones de un solo nivel de los problemas no convexos de optimización distribucionalmente robusta (DRO).
  • Ampliar los métodos existentes para manejar funciones simples multivariadas en DRO.
  • Permitir el cálculo de soluciones factibles y probadamente sólidas para una clase más amplia de problemas de DRO.

Principales métodos:

  • Aprovechando un enfoque de reformulación basado en la dualidad para problemas de DRO con funciones simples.
  • Extender el enfoque de parámetros de incertidumbre univariados a multivariados.
  • Implementación de una aproximación segura a través de una contraparte discretizada para restricciones duales semi-infinitas.
  • Formulación del problema como un programa semidefinido positivo de enteros mixtos computacionalmente tratable.

Principales resultados:

  • Se presenta una nueva aproximación segura para problemas DRO multivariados no convexos.
  • La aproximación permite la incorporación de información de momento y conjuntos de confianza en conjuntos de ambigüedad.
  • El método produce un programa semidefinido positivo de números enteros mixtos tratable que se puede resolver con el software existente.
  • La aproximación proporciona las condiciones suficientes para la robustez distributiva, garantizando soluciones probadamente robustas.

Conclusiones:

  • La aproximación segura propuesta amplía significativamente la aplicabilidad de las reformulaciones de DRO a problemas multivariados no convexos.
  • La tratabilidad algorítmica se logra a través de la discretización, lo que permite el cálculo práctico de soluciones robustas.
  • El método ofrece un enfoque computacionalmente eficiente y teóricamente sólido para resolver instancias complejas de DRO.