Worst Case Competitive Analysis of Online Algorithms for Conic Optimization

Eghbali, Reza; Fazel, Maryam

doi:10.48550/arxiv.1611.00507

Cited by 3 publications

(12 citation statements)

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“…A similar property is introduced in [11] and [12] as well and functions satisfying this property are called "smooth submodular" and "DR-submodular" there respectively. Additionally, [7] defined the DR property for concave functions with respect to a partial ordering induced by a cone and showed that by taking the cone to be R m + , Definition 2.1 is recovered and if the cone of positive semi-definite matrices is considered, the DR property generalizes to matrix ordering as well [13]. [12] showed that DR-submodular functions are concave along any non-negative direction, and any non-positive direction.…”

Section: Satisfies the Diminishing Returns (Dr) Property Ifmentioning

confidence: 99%

“…The Generalized Sequential algorithm reduces to the Sequential algorithm in [7] for K = 1 and hence the name. Additionally, this algorithm could be interpreted as the online counterpart of the offline Frank-Wolfe variant proposed in [12] for solving offline constrained continuous DR-submodular optimization problems.…”

Section: Algorithm 1 Generalized Sequential Algorithmmentioning

confidence: 99%

“…The definition above is inspired by the definition of α in [7]. The parameter α characterizes the curvature of the function.…”

Section: Competitive Ratio Analysismentioning

confidence: 99%

“…Online optimization covers a large number of problems including online resource allocation, online bipartite matching [1], the "Adwords" problem [2,3], online submodular welfare maximization [4], online linear programming [5] and online concave packing problem [6,7]. One type of algorithms proposed for solving such problems are primal-dual algorithms where the dual variable is updated at each step and is used to get the update rule for the primal variable [8].…”

Section: Introductionmentioning

confidence: 99%

“…For this setting, we obtain the first competitive ratio bound which is optimal in the special cases. Specifically, if the objective function is linear, our problem reduces to the online linear programming and the algorithm achieves the optimal competitive ratio [7,5]. If in addition to the linearity of the objective function, the linear packing constraint and the objective function are equal, the problem simplifies to the Adwords problem and we obtain the optimal 1 − 1 e competitive ratio [2,3] (note that since we allow fractional solutions for the Adwords problem, we do not need the small bids assumption to obtain the optimal competitive ratio).…”

Section: Introductionmentioning

confidence: 99%

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Competitive Algorithms for Online Budget-Constrained Continuous DR-Submodular Problems

Sadeghi,

Eghbali,

Fazel

2019

Preprint

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In this paper, we study a certain class of online optimization problems, where the goal is to maximize a function that is not necessarily concave and satisfies the Diminishing Returns (DR) property under budget constraints. We analyze a primal-dual algorithm, called the Generalized Sequential algorithm, and we obtain the first bound on the competitive ratio of online monotone DR-submodular function maximization subject to linear packing constraints which matches the known tight bound in the special case of linear objective function.Preprint. Under review.

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Section: Satisfies the Diminishing Returns (Dr) Property Ifmentioning

confidence: 99%

Section: Algorithm 1 Generalized Sequential Algorithmmentioning

confidence: 99%

“…The definition above is inspired by the definition of α in [7]. The parameter α characterizes the curvature of the function.…”

Section: Competitive Ratio Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Competitive Algorithms for Online Budget-Constrained Continuous DR-Submodular Problems

Sadeghi,

Eghbali,

Fazel

2019

Preprint

Self Cite

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Competitive online algorithms for resource allocation over the positive semidefinite cone

2018

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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.

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

Langbort

2018

Dyn Games Appl

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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.

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Worst Case Competitive Analysis of Online Algorithms for Conic Optimization

Cited by 3 publications

References 0 publications

Competitive Algorithms for Online Budget-Constrained Continuous DR-Submodular Problems

Competitive Algorithms for Online Budget-Constrained Continuous DR-Submodular Problems

Competitive online algorithms for resource allocation over the positive semidefinite cone

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

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