Convexification Techniques for Fractional Programs
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917694484250624 |
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| author | He, Taotao Liu, Siyue Tawarmalani, Mohit |
| author_facet | He, Taotao Liu, Siyue Tawarmalani, Mohit |
| contents | This paper develops a correspondence relating convex hulls of fractional functions with those of polynomial functions over the same domain. Using this result, we develop a number of new reformulations and relaxations for fractional programming problems. First, we relate 0-1 problems involving a ratio of affine functions with the boolean quadric polytope, and use inequalities for the latter to develop tighter formulations for the former. Second, we derive a new formulation to optimize a ratio of quadratic functions over a polytope using copositive programming. Third, we show that univariate fractional functions can be convexified using moment hulls. Fourth, we develop a new hierarchy of relaxations that converges finitely to the simultaneous convex hull of a collection of ratios of affine functions of 0-1 variables. Finally, we demonstrate theoretically and computationally that our techniques close a significant gap relative to state-of-the-art relaxations, require much less computational effort, and can solve larger problem instances. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_08424 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Convexification Techniques for Fractional Programs He, Taotao Liu, Siyue Tawarmalani, Mohit Optimization and Control This paper develops a correspondence relating convex hulls of fractional functions with those of polynomial functions over the same domain. Using this result, we develop a number of new reformulations and relaxations for fractional programming problems. First, we relate 0-1 problems involving a ratio of affine functions with the boolean quadric polytope, and use inequalities for the latter to develop tighter formulations for the former. Second, we derive a new formulation to optimize a ratio of quadratic functions over a polytope using copositive programming. Third, we show that univariate fractional functions can be convexified using moment hulls. Fourth, we develop a new hierarchy of relaxations that converges finitely to the simultaneous convex hull of a collection of ratios of affine functions of 0-1 variables. Finally, we demonstrate theoretically and computationally that our techniques close a significant gap relative to state-of-the-art relaxations, require much less computational effort, and can solve larger problem instances. |
| title | Convexification Techniques for Fractional Programs |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2310.08424 |