Bounds for approximate dynamic programming based on string optimization and curvature

Liu, Yajing; Chong, Edwin K. P.; Pezeshki, Ali; Moran, Bill

doi:10.1109/cdc.2014.7040433

Cited by 3 publications

(5 citation statements)

References 23 publications

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“…This extension introduces a number of issues that have to be addressed, which we do so here. Moreover, in the current paper, we introduce a new bounding result based on two curvature parameters, which is stronger than the bound used in [31]. This new bounding result is of interest in its own right.…”

Section: Prior Workmentioning

confidence: 94%

“…In our previous work [31], we had described bounding deterministic ADP schemes using a preliminary version of what is presented in this paper. The current paper goes well beyond [31] by treating the nontrivial extension to the stochastic case. This extension introduces a number of issues that have to be addressed, which we do so here.…”

Section: Prior Workmentioning

confidence: 99%

“…This model covers a wide variety of optimization problems found in many areas, ranging from engineering to economics. In particular, many adaptive sensing problems have this form (see, e.g., [31]).…”

Section: A Problem Formulationmentioning

confidence: 99%

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A General Framework for Bounding Approximate Dynamic Programming Schemes

Liu,

Chong,

Pezeshki

et al. 2018

Preprint

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We consider a broad family of control strategies called path-dependent action optimization (PDAO), where every control decision is treated as the solution to an optimization problem with a path-dependent objective function. How well such a scheme works depends on the chosen objective function to be optimized and, in general, it might be difficult to tell, without doing extensive simulation and testing, if a given PDAO design gives good performance or not. We develop a framework to bound the performance of PDAO schemes. We first introduce a general performance bound, in terms of two curvature parameters, for the greedy scheme in the string optimization problems under the condition that the objective function is prefix monotone. Then we show that every PDAO scheme is a greedy scheme for some optimization problem, and if that optimization problem is equivalent to our problem of interest and is provably prefix monotone, then we can say that our PDAO scheme is no worse than a certain factor of optimal. We show how to apply our framework to stochastic optimal control problems to bound the performance of approximate dynamic programming (ADP) schemes. Such schemes are based on approximating the expected value-togo term in Bellman's principle by computationally tractable means. Our framework provides the first systematic approach to bounding the performance of general ADP methods in the stochastic setting.

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Section: Prior Workmentioning

confidence: 94%

Section: Prior Workmentioning

confidence: 99%

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A General Framework for Bounding Approximate Dynamic Programming Schemes

Liu,

Chong,

Pezeshki

et al. 2018

Preprint

Self Cite

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“…In [36] and [37], Zhang et al generalized the notions of total curvature and elemental curvature to string submodular functions where the objective function value depends on the order of the elements in the set. This framework is further extended to approximate dynamic programming problems by Liu et al in [38].…”

Section: A Related Workmentioning

confidence: 99%

Subspace Selection for Projection Maximization With Matroid Constraints

Zhang

Wang

Chong

et al. 2017

IEEE Trans. Signal Process.

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Abstract-Suppose that there is a ground set which consists of a large number of vectors in a Hilbert space. Consider the problem of selecting a subset of the ground set such that the projection of a vector of interest onto the subspace spanned by the vectors in the chosen subset reaches the maximum norm. This problem is generally NP-hard, and alternative approximation algorithms such as forward regression and orthogonal matching pursuit have been proposed as heuristic approaches. In this paper, we investigate bounds on the performance of these algorithms by introducing the notions of elemental curvatures. More specifically, we derive lower bounds, as functions of these elemental curvatures, for performance of the aforementioned algorithms with respect to that of the optimal solution under uniform and non-uniform matroid constraints, respectively. We show that if the elements in the ground set are mutually orthogonal, then these algorithms are optimal when the matroid is uniform and they achieve at least 1/2-approximations of the optimal solution when the matroid is non-uniform.

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“…Therefore, it significantly reduces the search space, i.e., it does not search over all possible combinations of open expansion valves. When the expansion valve optimization problem (3.3) is extended to multi-stage optimization, a greedy algorithm achieves the same suboptimality bound using adaptive (or string) submodularity and monotonicity of the value function [8,2,15].…”

Section: Theorem 3 ([18]mentioning

confidence: 99%

Online Combinatorial Optimization for Interconnected Refrigeration Systems: Linear Approximation and Submodularity * *This work was supported in part by the NSF under CRII:CPS (CNS1657100).

Yang

2017

IFAC-PapersOnLine

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Commercial refrigeration systems consume 7% of the total commercial energy consumption in the United States. Improving their energy efficiency contributes to the sustainability of global energy systems and the supermarket business sector. This paper proposes a new control method that can save the energy consumption of multi-case supermarket refrigerators by explicitly taking into account their interconnected and switched system dynamics. Its novelty is a bilevel combinatorial optimization formulation to generate ON/OFF control actions for expansion valves and compressors. The inner optimization module keeps display case temperatures in a desirable range and the outer optimization module minimizes energy consumption. In addition to its energy-saving capability, the proposed controller significantly reduces the frequency of compressor switchings by employing a conservative compressor control strategy. However, solving this bilevel optimization problem associated with interconnected and switched systems is a computationally challenging task. To solve the problem in near real time, we propose two approximation algorithms that can solve both the inner and outer optimization problems at once. The first algorithm uses a linear approximation, and the second is based on the submodular structure of the optimization problem. Both are (polynomial-time) scalable algorithms and generate near-optimal solutions with performance guarantees. Our work complements existing optimization-based control methods (e.g., MPC) for supermarket refrigerators, as our algorithms can be adopted as a tool for solving combinatorial optimization problems arising in these methods.

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Bounds for approximate dynamic programming based on string optimization and curvature

Cited by 3 publications

References 23 publications

A General Framework for Bounding Approximate Dynamic Programming Schemes

A General Framework for Bounding Approximate Dynamic Programming Schemes

Subspace Selection for Projection Maximization With Matroid Constraints

Online Combinatorial Optimization for Interconnected Refrigeration Systems: Linear Approximation and Submodularity * *This work was supported in part by the NSF under CRII:CPS (CNS1657100).

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