Worst case competitive analysis for online conic optimization

Eghbali, Reza; Fazel, Maryam; Mesbahi, Mehran

doi:10.1109/cdc.2016.7798549

Cited by 10 publications

(11 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…• For general procurement cost functions, we write the surrogate function design problem as a quasiconvex optimization problem in which the optimization variables define the function. This strategy comes from adopting an optimization perspective for maximizing the competitive ratio similar to [EF16]. This technique allows us to construct surrogate functions for a wide class of procurement cost functions beyond those that are separable [HK18] and polynomial [CHK15].…”

Section: Contributionsmentioning

confidence: 99%

Function Design for Improved Competitive Ratio in Online Resource Allocation with Procurement Costs

Ray¹,

Sadeghi²,

Ratliff³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

We study the problem of online resource allocation, where multiple customers arrive sequentially and the seller must irrevocably allocate resources to each incoming customer while also facing a procurement cost for the total allocation. Assuming resource procurement follows an a priori known marginally increasing cost function, the objective is to maximize the reward obtained from fulfilling the customers' requests sans the cumulative procurement cost. We analyze the competitive ratio of a primal-dual algorithm in this setting, and develop an optimization framework for synthesizing a surrogate function for the procurement cost function to be used by the algorithm, in order to improve the competitive ratio of the primal-dual algorithm.Our first design method focuses on polynomial procurement cost functions and uses the optimal surrogate function to provide a more refined bound than the state of the art. Our second design method uses quasiconvex optimization to find optimal design parameters for a general class of procurement cost functions. Numerical examples are used to illustrate the design techniques. We conclude by extending the analysis to devise a posted pricing mechanism in which the algorithm does not require the customers' preferences to be revealed.

show abstract

Section: Contributionsmentioning

confidence: 99%

Function Design for Improved Competitive Ratio in Online Resource Allocation with Procurement Costs

Ray¹,

Sadeghi²,

Ratliff³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The following two lemmas are essentially taken from [EF16b]. We include proofs in Appendix A to make the present paper self-contained.…”

Section: Duality Gapmentioning

confidence: 99%

Competitive online algorithms for resource allocation over the positive semidefinite cone

2018

Self Cite

View full text Add to dashboard Cite

We consider a new and general online resource allocation problem, where the goal is to maximize a function of a positive semidefinite (PSD) matrix with a scalar budget constraint. The problem data arrives online, and the algorithm needs to make an irrevocable decision at each step. Of particular interest are classic experiment design problems in the online setting, with the algorithm deciding whether to allocate budget to each experiment as new experiments become available sequentially.We analyze two greedy primal-dual algorithms and provide bounds on their competitive ratios. Our analysis relies on a smooth surrogate of the objective function that needs to satisfy a new diminishing returns (PSD-DR) property (that its gradient is order-reversing with respect to the PSD cone). Using the representation for monotone maps on the PSD cone given by Löwner's theorem, we obtain a convex parametrization of the family of functions satisfying PSD-DR. We then formulate a convex optimization problem to directly optimize our competitive ratio bound over this set. This design problem can be solved offline before the data start arriving. The online algorithm that uses the designed smoothing is tailored to the given cost function, and enjoys a competitive ratio at least as good as our optimized bound. We provide examples of computing the smooth surrogate for D-optimal and A-optimal experiment design, and demonstrate the performance of the custom-designed algorithm.

show abstract

“…It is not necessary to run the shadow vertex simplex method every time player 2 adds a new action since player 2's new action may have no influence on player 1's security strategy and the game value. Comparing the old LP (1-4) with the new LP (5-9), we see that the new LP adds a new constraint (7) to the old LP. Geometrically, a new constraint means a new cut of the existing feasibility set.…”

Section: Player 1's Unchanging Security Strategymentioning

confidence: 99%

“…Online conic optimization is another online tool which looks for an approximated optimum of a convex optimization problem where new variables are introduced to the existing convex optimization [7]. A special requirement in online conic optimization is that the constraints of variable are separated.…”

mentioning

confidence: 99%

“…A special requirement in online conic optimization is that the constraints of variable are separated. Eghbali et.al provides an algorithm to approximate the optimum based on the form of the optimal primal-dual pair which is related to the concave conjugate of the objective function [7]. The main issue to adopt the algorithm in our problem is that to separate the constraints of variables, we need to reformulate the objective function, and the concave conjugate of the reformulated objective function is not well defined.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Iterative Computation of Security Strategies of Matrix Games with Growing Action Set

Langbort

2018

Dyn Games Appl

View full text Add to dashboard Cite

This paper studies how to efficiently update the saddle-point strategy, or security strategy of one player in a matrix game when the other player develops new actions in the game. It is well known that the saddle-point strategy of one player can be computed by solving a linear program. Developing a new action will add a new constraint to the existing LP. Therefore, our problem becomes how to solve the new LP with a new constraint efficiently. Considering the potentially huge number of constraints, which corresponds to the large size of the other player's action set, we use shadow vertex simplex method, whose computational complexity is lower than linear with respect to the size of the constraints, as the basis of our iterative algorithm. We first rebuild the main theorems in shadow vertex method with relaxed assumption to make sure such method works well in our model, then analyze the probability that the old optimum remains optimal in the new LP, and finally provides the iterative shadow vertex method whose computational complexity is shown to be strictly less than that of shadow vertex method. The simulation results demonstrates our main results about the probability of re-computing the optimum and the computational complexity of the iterative shadow vertex method.

show abstract

Worst case competitive analysis for online conic optimization

Cited by 10 publications

References 41 publications

Function Design for Improved Competitive Ratio in Online Resource Allocation with Procurement Costs

Function Design for Improved Competitive Ratio in Online Resource Allocation with Procurement Costs

Competitive online algorithms for resource allocation over the positive semidefinite cone

Iterative Computation of Security Strategies of Matrix Games with Growing Action Set

Contact Info

Product

Resources

About