A Practical Guide to Robust Optimization

Robust optimization is a young and active research field that has been mainly developed in the last 15 years. Robust optimization is very useful for practice, since it is tailored to the information at hand, and it leads to computationally tractable formulations. It is therefore remarkable that real-life applications of robust optimization are still lagging behind; there is much more potential for real-life applications than has been exploited hitherto. The aim of this paper is to help practitioners to understand robust optimization and to successfully apply it in practice. We provide a brief introduction to robust optimization, and also describe important do's and don'ts for using it in practice. We use many small examples to illustrate our discussions

A universal and structured way to derive dual optimization problem formulations

The dual problem of a convex optimization problem can be obtained in a relatively simple and structural way by using a well-known result in convex analysis, namely Fenchel’s duality theorem. This alternative way of forming a strong dual problem is the subject of this paper. We recall some standard results from convex analysis and then discuss how the dual problem can be written in terms of the conjugates of the objective function and the constraint functions. This is a didactically valuable method to explicitly write the dual problem. We demonstrate the method by deriving dual problems for several classical problems and also for a practical model for radiotherapy treatment planning, for which deriving the dual problem using other methods is a more tedious task. Additional material is presented in the appendices, including useful tables for finding conjugate functions of many functions

Thermodynamics of histories for the one-dimensional contact process

The dynamical activity K(t) of a stochastic process is the number of times it changes configuration up to time t. It was recently argued that (spin) glasses are at a first order dynamical transition where histories of low and high activity coexist. We study this transition in the one-dimensional contact process by weighting its histories by exp(sK(t)). We determine the phase diagram and the critical exponents of this model using a recently developed approach to the thermodynamics of histories that is based on the density matrix renormalisation group. We find that for every value of the infection rate, there is a phase transition at a critical value of s. Near the absorbing state phase transition of the contact process, the generating function of the activity shows a scaling behavior similar to that of the free energy in an equilibrium system near criticality.Comment: 16 pages, 7 figure

Effects of interactions between anthropogenic stressors and recurring perturbations on ecosystem resilience and collapse

Insights into declines in ecosystem resilience and their causes and effects can inform preemptive action to avoid ecosystem collapse and loss of biodiversity, ecosystem services, and human well-being. Empirical studies of ecosystem collapse are rare and hampered by ecosystem complexity, nonlinear and lagged responses, and interactions across scales. We investigated how an anthropogenic stressor could diminish ecosystem resilience to a recurring perturbation by altering a critical ecosystem driver. We studied groundwater-dependent, peat-accumulating, fire-prone wetlands known as upland swamps in southeastern Australia. We hypothesized that underground mining (stressor) reduces resilience of these wetlands to landscape fires (perturbation) by diminishing groundwater, a key ecosystem driver. We monitored soil moisture as an indicator of ecosystem resilience during and after underground mining. After landscape fire, we compared responses of multiple state variables representing ecosystem structure, composition, and function in swamps within the mining footprint with unmined reference swamps. Soil moisture declined without recovery in swamps with mine subsidence (i.e., undermined), but was maintained in reference swamps over 8 years (effect size 1.8). Relative to burned reference swamps, burned undermined swamps showed greater loss of peat via substrate combustion; reduced cover, height, and biomass of regenerating vegetation; reduced postfire plant species richness and abundance; altered plant species composition; increased mortality rates of woody plants; reduced postfire seedling recruitment; and extirpation of a hydrophilic animal. Undermined swamps therefore showed strong symptoms of postfire ecosystem collapse, whereas reference swamps regenerated vigorously. We found that an anthropogenic stressor diminished the resilience of an ecosystem to recurring perturbations, predisposing it to collapse. Avoidance of ecosystem collapse hinges on early diagnosis of mechanisms and preventative risk reduction. It may be possible to delay or ameliorate symptoms of collapse or to restore resilience, but the latter appears unlikely in our study system due to fundamental alteration of a critical ecosystem driver. Efectos de las interacciones entre los estresantes antropogénicos y las perturbaciones recurrentes sobre la resiliencia y el colapso de los ecosistemas