NON-ADDITIVE MEASURES

Pandey, Puja and Jayaram, Balasubramaniam (2018) NON-ADDITIVE MEASURES. Masters thesis, Indian Institute of Technology Hyderabad.

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Abstract

In the August-December 2017 semester, we studied the concept of non-additive measures (also known as Fuzzy Measures). We started by discussing classical measure. Classical Measures are nonnegative real-valued set functions, each de�ned on a speci�c class of subsets of a given universal set, that satisfy certain axiomatic requirements. One of these requirements, crucial to classical measures, is known as the requirement of additivity. Next, we de�ned monotone measures. Monotone measures are classi�ed into four classes, namely, additive measure, superadditive measure, subadditive measure and measures that do not belong to any of the three classes. Some non-additive measures such as Choquet Capacity, Sugeno measure, Belief and Plausibility Measures are also discussed and exempli�ed in this report. Finally, we discussed an application of belief and plausibility measure in decision making. In the January-May 2018 Semester, we study optimization of non-additive measures and as a �rst step we study optimization over additive measures, i.e., probability measures. We see that real life optimization problems contain uncertain data. These data can be uncertain due to measurement or estimation errors or implementation errors due to physical impossibility. There are two approaches to deal with these errors, which are Stochastic Optimization and Robust Optimization. We study robust optimization formulation for coherent risk function minimization. In this section, we also study conditions for convexifying the problem. Then, we study examples and applications of norm-constrained coherent risk minimization. Finally, we study portfolio optimization and binary Classi�cation using SVM.

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