Advancing Parametric Optimization

1. Front Matter

   Pages i-xii

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2. Introduction

   • Nathan Adelgren

   Pages 1-6

3. Background on mpLCP

   • Nathan Adelgren

   Pages 7-15

4. Algebraic Properties of Invariancy Regions

   • Nathan Adelgren

   Pages 17-30

5. Phase 2: Partitioning the Parameter Space

   • Nathan Adelgren

   Pages 31-65

6. Phase 1: Determining an Initial Feasible Solution

   • Nathan Adelgren

   Pages 67-78

7. Further Considerations

   • Nathan Adelgren

   Pages 79-87

8. Assessment of Performance

   • Nathan Adelgren

   Pages 89-93

9. Conclusion

   • Nathan Adelgren

   Pages 95-95

10. Back Matter

    Pages 97-113

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The theory presented in this work merges many concepts from mathematical optimization and real algebraic geometry. When unknown or uncertain data in an optimization problem is replaced with parameters, one obtains a multi-parametric optimization problem whose optimal solution comes in the form of a function of the parameters.The theory and methodology presented in this work allows one to solve both Linear Programs and convex Quadratic Programs containing parameters in any location within the problem data as well as multi-objective optimization problems with any number of convex quadratic or linear objectives and linear constraints. Applications of these classes of problems are extremely widespread, ranging from business and economics to chemical and environmental engineering. Prior to this work, no solution procedure existed for these general classes of problems except for the recently proposed algorithms

• multiparametric programming
• multiparametric linear programming
• multiparametric quadratic programming
• multiobjective quadratic programming
• semi-algebraic sets
• polynomial optimization

• Department of Mathematics and Computer Science, Edinboro University, Edinboro, USA

  Nathan Adelgren

​Nathan Adelgren earned his Ph.D. in Mathematical Sciences from Clemson University in 2016. He is currently an Associate Professor in the Department of Mathematics and Computer Science at Edinboro University in Edinboro, PA. His research interests are in the general field of Operations Research and include developing novel solution procedures for nontraditional optimization problems in the form of multicriteria, multiparametric, and mixed-integer programs as well as various combinations of these.

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