We consider smooth stochastic convex optimization problems in the context of algorithms which are based on directional derivatives of the objective function. This context can be considered as an intermediate one between derivative-free optimization and gradient-based optimization. We assume that at any given point and for any given direction, a stochastic approximation for the directional derivative of the objective function at this point and in this direction is available with some additive noise. The noise is assumed to be of an unknown nature, but bounded in the absolute value. We underline that we consider directional derivatives in any direction, as opposed to coordinate descent methods which use only derivatives in coordinate directions. For this setting, we propose a non-accelerated and an accelerated directional derivative method and provide their complexity bounds. Our non-accelerated algorithm has a complexity bound which is similar to the gradient-based algorithm, that is, without any dimension-dependent factor. Our accelerated algorithm has a complexity bound which coincides with the complexity bound of the accelerated gradient-based algorithm up to a factor of square root of the problem dimension. We extend these results to strongly convex problems.
We consider the ranking of decision alternatives in decision analysis problems under uncertainty, under very weak assumptions about the type of utility function and information about the probabilities of the states of nature. Namely, the following two assumptions are required for the suggested method: the utility function is in the class of increasing continuous functions, and the probabilities of the states of nature are rank-ordered. We develop a simple analytical method for the partial ranking of decision alternatives under the stated assumptions. This method does not require solving optimization programs and is free of the rounding errors.
We introduce the notion of a maximum likelihood optimal decision alternative for the choice problem with a finite set of decision alternatives, assuming a general parametric preference model of the decision maker. We also develop an optimization-based method for the identification of such alternatives for the cases in which the parametric preference model is based on uncertain intervals for criterion trade-offs. The suggested approach can be seen as generalising stochastic multicriteria acceptability analysis (SMAA) to a wider modelling setting. It also provides a maximum likelihood interpretation of the SMAA acceptability index.
In this paper we obtain new theoretical results relating the notions of potential optimality and non-dominance without an assumption that a value function exists. In particular, we investigate a decision problem involving the choice of single or multiple best objects. Our results show that the notions of potential optimality and non-dominance are equivalent in a special case of preferences of the decision maker expressed by partial quasi-orders.
We study questions of robustness of linear multiple objective problems in the sense of post-optimal analysis, that is, we study conditions under which a given efficient solution remains efficient when the criteria/objective matrix undergoes some alterations. We consider addition or removal of certain criteria, convex combination with another criteria matrix, or small perturbations of its entries. We provide a necessary and sufficient condition for robustness in a verifiable form and give two formulae to compute the radius of robustness.
In this paper we consider choice problems under the assumption that the preferences of the decision maker are expressed in the form of a parametric partial weak order without assuming the existence of any value function. We investigate both the sensitivity (stability) of each non-dominated solution with respect to the changes of parameters of this order, and the sensitivity of the set of non-dominated solutions as a whole to similar changes. We show that this type of sensitivity analysis can be performed by employing techniques of linear programming.
This paper investigates the coordinated scheduling problem of production and transportation in a two-stage supply chain, where the actual job processing time is a linear function of its starting time. During the production stage the jobs are first processed in serial batches on a bounded serial batching machine at the manufacturer's site. Then, the batches are delivered to a customer by a single vehicle with limited capacity during the transportation stage, and the vehicle can only deliver one batch at one time. The objective of this proposed scheduling problem is to make decisions on job batching and batch sequencing so as to minimize the makespan. Moreover, we consider two different models. With regards to the scheduling model with a buffer for storing the processed batches before transportation, we develop an optimal algorithm to solve it. For the scheduling model without buffer, we present some useful properties and develop a heuristic H for solving it. Then a novel lower bound is derived and two optimal algorithms are designed for solving two special cases. Furthermore, computational experiments with random instances of different sizes are conducted to evaluate the proposed heuristic H, and the results show that our proposed algorithm is superior to other four approaches in the literature. Besides, heuristic H in our experiments can effectively and efficiently solve both small-size and large-size problems in a reasonable time.
We consider the problem of selecting a predetermined number of objects from a given finite set. It is assumed that the preferences of the decisionmaker on this set are only partially known. Our solution approach is based on the notions of optimal and non-dominated subsets. The properties of such subsets and the objects they contain are investigated. The implementation of the developed approach is discussed and illustrated by various examples.
This paper analyses in a systematic way the effect of benefit and cost parameter heterogeneity on the stability and effectiveness of International Environmental Agreements. Compared to existing literature, we consider a more general form of heterogeneity, different functional forms, and alternative collective decision making processes in coalitions with and without transfers. Using systematic numerical simulations and novel visualization techniques, we show that transfers are crucial to overcome heterogeneity both in terms of stability and effectiveness. Without, or with limited transfers heterogeneous coalitions are often unstable and ineffective. More research on less than ideal transfer schemes and collective decision-making rules is necessary to bridge the gap between theoretical models and the reality of international negotiations on transboundary pollution problems.
The main goal of this paper is to model the effects of wholesale price control on manufacturer’s profit, taking explicitly into account the retailer’s sales motivation and performance. We consider a stylized distribution channel where a manufacturer sells a single kind of good to a single retailer. Wholesale price discounts are assumed to increase the retailer’s motivation thus improving sales. We study the manufacturer’s profit maximization problem as an optimal control model where the manufacturer’s control is the discount on wholesale price and retailer’s motivation is one of the state variables. In particular in the paper we prove that an increasing discount policy is optimal for the manufacturer when the retailer is not efficient while efficient retailers may require to decrease the trade discounts at the end of the selling period. Computational experiments point out how the discount on wholesale price passed by the retailer to the market (pass-through) influences the optimal profit of the manufacturer.
Wildfires are a common phenomenon on most continents. They have occurred for an estimated 60 mil- lion years and are part of a regular climatic cycle. Nevertheless, wildfires represent a real and continuing problem that can have a major impact on people, wildlife and the environment. The intensity and sever- ity of wildfires can be reduced through fuel management activities. The most common and effective fuel management activity is prescribed burning. We propose a multi-period optimization framework based on mixed integer programming (MIP) techniques to determine the optimal spatial allocation of prescribed burning activities over a finite planning horizon. In contrast to the existing fuel management optimiza- tion literature, we model fuel accumulation with Olson’s equation. To capture potential fire spread along with irregular landscape connectivity considerations, we use a graph-theoretical approach that allows us to exploit graph connectivity measures (e.g., the number of connected components) as optimization ob- jectives. The resulting mathematical programs can be tackled by general purpose MIP solvers, while for handling larger instances we propose a simple heuristic. Our computational experiments with test in- stances constructed based on real-life data reveal interesting insights and demonstrate the advantages and limitations of the proposed approaches.