A Model of Anticipated Regret and Endogenous Beliefs

This paper clarifies and extends the model of anticipated regret and endogenous beliefs based on the Savage (1951) Minmax Regret Criterion developped in Suryanarayanan (2006a). A decision maker chooses an action with state contingent consequences but cannot precisely assess the true probability distribution of the state. She distrusts her prior about the true distribution and surrounds it with a set of alternative but plausible probability distributions. The decision maker minimizes the worst expected regret over all plausible probability distributions and alternative actions, where regret is the loss experienced when the decision maker compares an action to a counterfactual feasible alternative for a given realization of the state. Preliminary theoretical results provide a systematic algorithm to find the solution to the decision problem and show how models of Minmax Regret differs from models of ambiguity aversion and expected utility. In particular, the solution to the decision problem can always be represented as a saddle point solution to an equivalent zerosum game problem. This new problem jointly produces the solution to the Anticipated Regret problem and the endogenous belief. We then use the endogenous belief to define the implicit certainty equivalent and to build an infinite horizon and time consistent problem for a decision maker minimizing her lifetime worst expected regrets.

Content Popularity Prediction Towards Location-Aware Mobile Edge Caching

Mobile edge caching enables content delivery within the radio access network, which effectively alleviates the backhaul burden and reduces response time. To fully exploit edge storage resources, the most popular contents should be identified and cached. Observing that user demands on certain contents vary greatly at different locations, this paper devises location-customized caching schemes to maximize the total content hit rate. Specifically, a linear model is used to estimate the future content hit rate. For the case where the model noise is zero-mean, a ridge regression based online algorithm with positive perturbation is proposed. Regret analysis indicates that the proposed algorithm asymptotically approaches the optimal caching strategy in the long run. When the noise structure is unknown, an $H_{\infty}$ filter based online algorithm is further proposed by taking a prescribed threshold as input, which guarantees prediction accuracy even under the worst-case noise process. Both online algorithms require no training phases, and hence are robust to the time-varying user demands. The underlying causes of estimation errors of both algorithms are numerically analyzed. Moreover, extensive experiments on real world dataset are conducted to validate the applicability of the proposed algorithms. It is demonstrated that those algorithms can be applied to scenarios with different noise features, and are able to make adaptive caching decisions, achieving content hit rate that is comparable to that via the hindsight optimal strategy.Comment: to appear in IEEE Trans. Multimedi

Ellipsoidal Prediction Regions for Multivariate Uncertainty Characterization

While substantial advances are observed in probabilistic forecasting for power system operation and electricity market applications, most approaches are still developed in a univariate framework. This prevents from informing about the interdependence structure among locations, lead times and variables of interest. Such dependencies are key in a large share of operational problems involving renewable power generation, load and electricity prices for instance. The few methods that account for dependencies translate to sampling scenarios based on given marginals and dependence structures. However, for classes of decision-making problems based on robust, interval chance-constrained optimization, necessary inputs take the form of polyhedra or ellipsoids. Consequently, we propose a systematic framework to readily generate and evaluate ellipsoidal prediction regions, with predefined probability and minimum volume. A skill score is proposed for quantitative assessment of the quality of prediction ellipsoids. A set of experiments is used to illustrate the discrimination ability of the proposed scoring rule for misspecification of ellipsoidal prediction regions. Application results based on three datasets with wind, PV power and electricity prices, allow us to assess the skill of the resulting ellipsoidal prediction regions, in terms of calibration, sharpness and overall skill.Comment: 8 pages, 7 Figures, Submitted to IEEE Transactions on Power System

Estudio de problemas de clasificación supervisada y de localización en redes mediante optimización matemática

This PhD dissertation addresses several problems in the fields of Supervised Classification and Location Theory using tools and techniques coming from Mathematical Optimization. A brief description of these problems and the methodologies proposed for their analysis and resolution is given below. In the first chapter, the principles of Supervised Classification and Location Theory are discussed in detail, emphasizing the topics studied in this thesis. The following two chapters discuss Supervised Classification problems. In particular, Chapter 2 proposes exact solution approaches for various models of Support Vector Machines (SVM) with ramp loss, a well-known classification method that limits the influence of outliers. The resulting models are analyzed to obtain initial bounds of the big M parameters included in the formulation. Then, solution approaches based on three strategies for obtaining tighter values of the big M parameters are proposed. Two of them require solving a sequence of continuous optimization problems, while the third uses the Lagrangian relaxation. The derived resolution methods are valid for the l1-norm and l2-norm ramp loss formulations. They are tested and compared with existing solution methods in simulated and real-life datasets, showing the efficiency of the developed methodology. Chapter 3 presents a new SVM-based classifier that simultaneously deals with the limitation of the influence of outliers and feature selection. The influence of outliers is taken under control using the ramp loss margin error criterion, while the feature selection process is carried out including a new family of binary variables and several constraints. The resulting model is formulated as a mixed-integer program with big M parameters. The characteristics of the model are analyzed and two different solution approaches (exact and heuristic) are proposed. The performance of the obtained classifier is compared with several classical ones in different datasets. The next two chapters deal with location problems, in particular, two variants of the Maximal Covering Location Problem (MCLP) in networks. These variants respond to the modeling of two different scenarios, with and without uncertainty in the input data. First, Chapter 4 presents the upgrading version of MCLP with edge length modifications on networks. This problem aims at locating p facilities on the nodes (of the network) so as to maximize coverage, considering that the length of the edges can be reduced within a budget. Hence, we have to decide on: the optimal location of p facilities and the optimal edge length reductions. To solve it, we propose three different mixed-integer formulations and a preprocessing phase for fixing variables and removing some constraints. Moreover, we analyze the characteristics of these formulations to strengthen them by proposing valid inequalities. Finally, we compare the three formulations and their corresponding improvements by testing their performance over different datasets. The following chapter, Chapter 5, also considers a MCLP, albeit from the perspective of uncertainty. In particular, this chapter addresses a version of the single-facility MCLP on a network where the demand is distributed along the edges and uncertain with only a known interval estimation. We propose a minmax regret model where the service facility can be located anywhere along the network. Furthermore, we present two polynomial algorithms for finding the location that minimizes the maximal regret assuming that the demand realization is an unknown constant or linear function on each edge. We also include two illustrative examples as well as a computational study to show the potential of the proposed methodology

Dynamics of Inductive Inference in a Unified Framework

We present a model of inductive inference that includes, as special cases, Bayesian reasoning, case-based reasoning, and rule-based reasoning. This unified framework allows us to examine, positively or normatively, how the various modes of inductive inference can be combined and how their relative weights change endogenously. We establish conditions under which an agent who does not know the structure of the data generating process will decrease, over the course of her reasoning, the weight of credence put on Bayesian vs. non-Bayesian reasoning. We show that even random data can make certain theories seem plausible and hence increase the weight of rule-based vs. case-based reasoning, leading the agent in some cases to cycle between being rule-based and case-based. We identify conditions under which minmax regret criteria will not be effective.Induction, Bayesian updating, Case-Based Reasoning, Inference

Distributionally Robust Optimization: A Review

The concepts of risk-aversion, chance-constrained optimization, and robust optimization have developed significantly over the last decade. Statistical learning community has also witnessed a rapid theoretical and applied growth by relying on these concepts. A modeling framework, called distributionally robust optimization (DRO), has recently received significant attention in both the operations research and statistical learning communities. This paper surveys main concepts and contributions to DRO, and its relationships with robust optimization, risk-aversion, chance-constrained optimization, and function regularization

Models and algorithms for deterministic and robust discrete time/cost trade-off problems

Ankara : The Department of Management, Bilkent University, 2008.Thesis (Ph.D.) -- Bilkent University, 2008.Includes bibliographical references leaves 136-145Projects are subject to various sources of uncertainties that often negatively impact activity durations and costs. Therefore, it is of crucial importance to develop effective approaches to generate robust project schedules that are less vulnerable to disruptions caused by uncontrollable factors. This dissertation concentrates on robust scheduling in project environments; specifically, we address the discrete time/cost trade-off problem (DTCTP). Firstly, Benders Decomposition based exact algorithms to solve the deadline and the budget versions of the deterministic DTCTP of realistic sizes are proposed. We have included several features to accelerate the convergence and solve large instances to optimality. Secondly, we incorporate uncertainty in activity costs. We formulate robust DTCTP using three alternative models. We develop exact and heuristic algorithms to solve the robust models in which uncertainty is modeled via interval costs. The main contribution is the incorporation of uncertainty into a practically relevant project scheduling problem and developing problem specific solution approaches. To the best of our knowledge, this research is the first application of robust optimization to DTCTP. Finally, we introduce some surrogate measures that aim at providing an accurate estimate of the schedule robustness. The pertinence of proposed measures is assessed through computational experiments. Using the insight revealed by the computational study, we propose a two-stage robust scheduling algorithm. Furthermore, we provide evidence that the proposed approach can be extended to solve a scheduling problem with tardiness penalties and earliness rewards.Hazır, ÖncüPh.D