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In this paper, we solve the multiple product price optimization problem under interval uncertainties of the price sensitivity parameters in the demand function. The objective of the price optimization problem is to maximize the overall revenue of the firm where the decision variables are the prices of the products supplied by the firm. We propose an approach that yields optimal solutions under different variations of the estimated price sensitivity parameters. We adopt a robust optimization approach by building a data-driven uncertainty set for the parameters, and then construct a deterministic counterpart for the robust optimization model. The numerical results show that two objectives are fulfilled: the method reflects the uncertainty embedded in parameter estimations, and also an interval is obtained for optimal prices. We also conducted a simulation study to which we compared the results of our approach. The comparisons show that although robust optimization is deemed to be conservative, the results of the proposed approach show little loss compared to those from the simulation.
To ensure a successful bid while maximizing of profits, generation companies (GENCOs) need a self-scheduling strategy that can cope with a variety of scenarios. So distributionally robust opti-mization (DRO) is a good choice because that it can provide an adjustable self-scheduling strategy for GENCOs in the uncertain environment, which can well balance robustness and economics compared to strategies derived from robust optimization (RO) and stochastic programming (SO). In this paper, a novel mo-ment-based DRO model with conditional value-at-risk (CVaR) is proposed to solve the self-scheduling problem under electricity price uncertainty. Such DRO models are usually translated into semi-definite programming (SDP) for solution, however, solving large-scale SDP needs a lot of computational time and resources. For this shortcoming, two effective approximate models are pro-posed: one approximate model based on vector splitting and an-other based on alternate direction multiplier method (ADMM), both can greatly reduce the calculation time and resources, and the second approximate model only needs the information of the current area in each step of the solution and thus information private is guaranteed. Simulations of three IEEE test systems are conducted to demonstrate the correctness and effectiveness of the proposed DRO model and two approximate models.
We consider optimization problems with uncertain constraints that need to be satisfied probabilistically. When data are available, a common method to obtain feasible solutions for such problems is to impose sampled constraints, following the so-called scenario optimization approach. However, when the data size is small, the sampled constraints may not statistically support a feasibility guarantee on the obtained solution. This paper studies how to leverage parametric information and the power of Monte Carlo simulation to obtain feasible solutions for small-data situations. Our approach makes use of a distributionally robust optimization (DRO) formulation that translates the data size requirement into a Monte Carlo sample size requirement drawn from what we call a generating distribution. We show that, while the optimal choice of this generating distribution is the one eliciting the data or the baseline distribution in a nonparametric divergence-based DRO, it is not necessarily so in the parametric case. Correspondingly, we develop procedures to obtain generating distributions that improve upon these basic choices. We support our findings with several numerical examples.
Setting an effective reserve price for strategic bidders in repeated auctions is a central question in online advertising. In this paper, we investigate how to set an anonymous reserve price in repeated auctions based on historical bids in a way that balances revenue and incentives to misreport. We propose two simple and computationally efficient methods to set reserve prices based on the notion of a clearing price and make them robust to bidder misreports. The first approach adds random noise to the reserve price, drawing on techniques from differential privacy. The second method applies a smoothing technique by adding noise to the training bids used to compute the reserve price. We provide theoretical guarantees on the trade-offs between the revenue performance and bid-shading incentives of these two mechanisms. Finally, we empirically evaluate our mechanisms on synthetic data to validate our theoretical findings.
In this paper, we study a retailer price optimization problem which includes the practical constraints: maximum number of price changes and minimum amount of price change (if a change is recommended). We provide a closed-form formula for the Euclidean projection onto the feasible set defined by these two constraints, based on which a simple gradient projection algorithm is proposed to solve the price optimization problem. We study the convergence and solution quality of the proposed algorithm. We extend the base model to include upper/lower bounds on the individual product prices and solve it with some adjustments to the gradient projection algorithm. Numerical results are reported to demonstrate the performance of the proposed algorithm.
Compressed sensing (CS) shows that a signal having a sparse or compressible representation can be recovered from a small set of linear measurements. In classical CS theory, the sampling matrix and representation matrix are assumed to be known exactly in advance. However, uncertainties exist due to sampling distortion, finite grids of the parameter space of dictionary, etc. In this paper, we take a generalized sparse signal model, which simultaneously considers the sampling and representation matrix uncertainties. Based on the new signal model, a new optimization model for robust sparse signal reconstruction is proposed. This optimization model can be deduced with stochastic robust approximation analysis. Both convex relaxation and greedy algorithms are used to solve the optimization problem. For the convex relaxation method, a sufficient condition for recovery by convex relaxation is given; For the greedy algorithm, it is realized by the introduction of a pre-processing of the sensing matrix and the measurements. In numerical experiments, both simulated data and real-life ECG data based results show that the proposed method has a better performance than the current methods.