Basic Properties of Convex Polytopes

Henk, Martin; Richter-Gebert, Jürgen; Ziegler, Günter M.

doi:10.1201/9781420035315.pt2

Cited by 70 publications

(92 citation statements)

References 19 publications

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“…This result follows directly from McMullen's upper bound theorem for facet enumeration (McMullen, 1970;Henk et al, 1997).…”

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confidence: 61%

Multi-objective decision-theoretic planning

Roijers

2016

AI Matters

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Decision making is hard. It o en requires reasoning about uncertain environments, partial observability and action spaces that are too large to enumerate. In such complex decisionmaking tasks decision-theoretic agents, that can reason about their environments on the basis of mathematical models and produce policies that optimize the utility for their users, can o en assist us.In most research on decision-theoretic agents, the desirability of actions and their e ects is codi ed in a scalar reward function. However, many real-world decision problems have multiple objectives. In such cases the problem is more naturally expressed using a vector-valued reward function. Rather than having a single optimal policy, we then want to produce a set of policies that covers all possible preferences between the objectives. We call such a set a coverage set. In this dissertation, we focus on decision-theoretic planning algorithms that produce the convex coverage set (CCS), which is the optimal solution set when either: 1) the user utility can be expressed as a weighted sum over the values for each objective; or 2) policies can be stochastic.We propose new methods based on two popular approaches to creating planning algorithms that produce an (approximate) CCS by building on an existing single-objective algorithm. In the inner loop approach, we replace the summations and maximizations in the inner most loops of the single-objective algorithm by cross-sums and pruning operations. In the outer loop approach, we solve a multi-objective problem as a series of scalarized problems by employing the single-objective method as a subroutine.Our most important contribution is an outer loop framework that we call optimistic linear support (OLS). As an outer loop method OLS builds the CCS incrementally. We show that, contrary to existing outer loop methods, each intermediate result is a bounded approximation of the CCS with known bounds (even when the single-objective method used is a bounded approximate method as well) and is guaranteed to terminate in a nite number of iterations.We apply OLS-based algorithms to a variety of multi-objective decision problems, and show that it is more memory-e cient, and faster than corresponding inner loop algorithms for moderate numbers of objectives. We show that exchanging subroutines in OLS is relatively easy and illustrate the importance on a complex planning problem. Finally, we show that it is o en possible to reuse parts of the policies and values, found in earlier iterations of OLS, to hot-start later iterations of OLS. Using this last insight, we propose the rst method for multi-objective POMDPs that employs point-based planning and can produce an ε-CCS in reasonable time.Overall, the methods we propose bring us closer to truly practical multi-objective decisiontheoretic planning.

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“…This result follows directly from McMullen's upper bound theorem for facet enumeration (McMullen, 1970;Henk et al, 1997).…”

mentioning

confidence: 61%

Multi-objective decision-theoretic planning

Roijers

2016

AI Matters

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show abstract

“…Формирование комбинаторных мно-гогранников -это основная задача, решаемая в поли-эдральной комбинаторике [11,12]. На сегодняшний день она решена для таких евклидовых комбинаторных множеств как множество перестановок [1,3,13] и его обобщений, таких как общие перестановочное и поли-перестановочное множества [1,2,14]; для общих мно-жеств размещений и полиразмещений [2], отдельных классов сочетаний [7], некоторых 0-1-множеств [5] та-ких как множество перестановочных матриц [6] и т. п.…”

Section: анализ литературных данных и постановка задачиunclassified

“…Мультиповерхностный подход состоит в форми-ровании дискретного множества из строгой части (11) функционального представления с последующим от-сечением, в случае необходимости, недопустимых то-чек при помощи ограничений нестрогой части (13).…”

Section: подходы к построению функциональных представлений множествunclassified

“…Продемонстрируем применимость перечисленных подходов к построению функциональных представле-ний и возможность с их помощью построения строгих функциональных представлений вида (10) на обобще-нии множества перестановок [1,3,13] -общем переста-новочном множестве [1,2,14].…”

Section: подходы к построению функциональных представлений множествunclassified

“…1. Одно функциональное представления n E Частным случаем nk E (G) является множество n n n E E (J ) = перестановок из первых n натуральных чисел [1,3,13], для которого в [15] предложено следую-щее функциональное представление:…”

Section: непрерывные представления общих перестановокunclassified

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This paper deals with the construction of seismic response interaction diagrams that show the correlation of multiple responses and are important to determine the critical combination of modal responses. Many design problems, such as column design under combined axial force and bending moments, fall into this category. We address general modal and multicomponent combination rules and study their effect on the shape of the response interaction diagrams, thus extending previous work done for quadratic combination rules. Special attention is given to multilinear combination rules which lead to polyhedral shapes. Having developed efficient methods to deal with polyhedral shapes, we explore the idea of adopting a multilinear modal combination rule to compose with a multicomponent percentage rule.

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Basic Properties of Convex Polytopes

Cited by 70 publications

References 19 publications

Multi-objective decision-theoretic planning

Multi-objective decision-theoretic planning

Funcrional and analytic representations of the total permutation

The shape of the seismic response interaction diagram: The effect of combination rules in response spectrum analysis

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