Robust optimization of contaminant sensor placement for community water systems

We present a series of related robust optimization models for placing sensors in municipal water networks to detect contaminants that are maliciously or accidentally injected. We formulate sensor placement problems as mixed-integer programs, for which the objective coefficients are not known with certainty. We consider a restricted absolute robustness criteria that is motivated by natural restrictions on the uncertain data, and we define three robust optimization models that differ in how the coefficients in the objective vary. Under one set of assumptions there exists a sensor placement that is optimal for all admissible realizations of the coefficients. Under other assumptions, we can apply sorting to solve each worst-case realization efficiently, or we can apply duality to integrate the worst-case outcome and have one integer program. The most difficult case is where the objective parameters are bilinear, and we prove its complexity is NP-hard even under simplifying assumptions. We consider a relaxation that provides an approximation, giving an overall guarantee of near-optimality when used with branch-and-bound search. We present preliminary computational experiments that illustrate the computational complexity of solving these robust formulations on sensor placement applications.     

Approximation for the mean value performance of locking algorithms for distributed database systems: a partitioned database

Concurrent access to databases must be synchronized for correct execution of transactions and preservation of data consistency. This is usually achieved through use of concurrency control algorithms, amongst which locking algorithms are the most popular both in the literature and in practice. Several analytic methods have been developed for predicting the performance of centralized database systems employing locking algorithms for concurrency control, but very few exist for distributed database systems.This paper proposes a method to approximate the mean value of various performance parameters in distributed database systems using locking for concurrency control. The main contribution of this approach is its ability to model the interaction between resource and data contention and the resulting effect on system performance. System performance is evaluated at a point where the interaction between these two factors is in equilibrium (stable state) and both the data and resource contention equations are simultaneously satisfied.The model involves the solution of a set of simultaneous polynomial equations whose order is dependent on several problem parameters such as the number of nodes and number of locks requested per transaction. These equations are solved by an iterative procedure to evaluate approximate values of relative throughput, utilization of servers and transaction response time. The small computational requirements of the analytical model permit sensitivity analysis on network parameters, and can thus be effectively used by system designers to evaluate choices of communication line speeds, processor capacity, database sizes, etc.The analytic approximations have been extensively verified against simulations for networks with up to 20 nodes. The input traffic was varied from light loads (about 5% utilization of the channels and processors) to heavy loads (about 65% utilization of the processors and channels). The discrepancies between the analytic approximation and the simulation were quite small (2–8%).This work was done while the authors were at Drexel University, Philadelphia.     

Robust optimization approximation for joint chance constrained optimization problem

Chance constraint is widely used for modeling solution reliability in optimization problems with uncertainty. Due to the difficulties in checking the feasibility of the probabilistic constraint and the non-convexity of the feasible region, chance constrained problems are generally solved through approximations. Joint chance constrained problem enforces that several constraints are satisfied simultaneously and it is more complicated than individual chance constrained problem. This work investigates the tractable robust optimization approximation framework for solving the joint chance constrained problem. Various robust counterpart optimization formulations are derived based on different types of uncertainty set. To improve the quality of robust optimization approximation, a two-layer algorithm is proposed. The inner layer optimizes over the size of the uncertainty set, and the outer layer optimizes over the parameter t which is used for the indicator function upper bounding. Numerical studies demonstrate that the proposed method can lead to solutions close to the true solution of a joint chance constrained problem.     

Robust inverse optimization

Given an observation of a decision-maker’s uncertain behavior, we develop a robust inverse optimization model for imputing an objective function that is robust against mis-specifications of the behavior. We characterize the inversely optimized cost vectors for uncertainty sets that may or may not intersect the feasible region, and propose tractable solution methods for special cases. We demonstrate the proposed model in the context of diet recommendation.     

Robust ℋ∞ performance analysis and synthesis of discrete-time LPV systems

We propose a less conservative linear matrix inequalities (LMIs) condition for ?_∞ performance analysis and synthesis of discrete-time linear parameter varying (LPV) systems. The time varying parameter is represented by a linear fractional form. Sufficient conditions for ?_∞ performance and controller synthesis are given in terms of a finite number of LMIs using parameter dependent Lyapunov functions.     

Robust optimization for routing problems on trees

The prize-collecting travelling salesman problem (pc-tsp) is a known variant of the classical travelling salesman problem. In the pc-tsp, we are given a weighted graph \(G=(V,E)\) with edge weights \(\ell :E\rightarrow {\mathbb {R}}_+\), a special vertex \(r\in V\), penalties \(\pi :V\rightarrow {\mathbb {R}}_+\) and the goal is to find a closed tour K such that \(r\in V(K)\) and such that the cost \(\ell (K)+\pi (V{\setminus } V(K))\), which is the sum of the weights of the edges in the tour and the cost of the vertices not spanned by K, is minimized. In this paper, we study the pc-tsp from a viewpoint of robust optimization. In our setting, an operator must find a closed tour in a (known) edge-weighted tree \(T=(V,E)\) starting and ending in the depot r while some of the edges may be blocked by “avalanches” defining the scenario \(\xi \). The cost \(f(K,\xi )\) of a tour K in scenario \(\xi \) is the cost resulting from “shortcutting” K, i.e. from restricting K to the nodes which are reachable from r in scenario \(\xi \), i.e. in the graph \(T {\setminus } \xi =(V,E{\setminus }\xi )\). In the absolute robust version of the problem one searches for a tour which minimizes the worst-case cost over all scenarios, while in the deviation robust counterpart, the regret, that is, the deviation from an optimum solution for a particular scenario, is sought to be minimized. We show that both versions of the problem can be solved in polynomial time on trees.     

Robust optimization of uncertain multistage inventory systems with inexact data in decision rules

In production-inventory problems customer demand is often subject to uncertainty. Therefore, it is challenging to design production plans that satisfy both demand and a set of constraints on e.g. production capacity and required inventory levels. Adjustable robust optimization (ARO) is a technique to solve these dynamic (multistage) production-inventory problems. In ARO, the decision in each stage is a function of the data on the realizations of the uncertain demand gathered from the previous periods. These data, however, are often inaccurate; there is much evidence in the information management literature that data quality in inventory systems is often poor. Reliance on data “as is” may then lead to poor performance of “data-driven” methods such as ARO. In this paper, we remedy this weakness of ARO by introducing a model that treats past data itself as an uncertain model parameter. We show that computational tractability of the robust counterparts associated with this extension of ARO is still maintained. The benefits of the new model are demonstrated by a numerical test case of a well-studied production-inventory problem. Our approach is also applicable to other ARO models outside the realm of production-inventory planning.     

Robust optimization models for managing callable bond portfolios

A major sector of the bond markets is currently represented by instruments with embedded call options. The complexity of bonds with call features, coupled with the recent increase in volatility, has raised the risks as well as the potential rewards for bond holders. These complexities, however, make it difficult for the portfolio manager to evaluate individual securities and their associated risks in order to successfully construct bond portfolios. Traditional bond portfolio management methods are inadequate, particularly when interest-rate-dependent cashflows are involved. In this paper we integrate traditional simulation models for bond pricing with recent developments in robust optimization to develop tools for the management of portfolios of callable bonds. Two models are developed: a single-period model that imposes robustness by penalizing downside tracking error, and a multi-stage stochastic program with recourse. Both models are applied to create a portfolio to track a callable bond index. The models are backtested using ex poste market data over the period from January 1992 to March 1993, and they perform constistently well.     

Robust inversion algorithms for vector linear systems

We consider the inversion problem for linear systems, which involves estimation of the unknown input vector. The inversion problem is considered for a system with a vector output and a vector input assuming that the observed output is of higher dimension than the unknown input. The problem is solved by using a controlled model in which the control stabilizes the deviations of the model output from the system output. The stabilizing model control or its averaged form may be used as the estimate of the unknown system input. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 17–22, 2004.     

Robust analysis and global optimization

In this paper, the properties of robust sets and robust functions are studied. Also, we study minimization of a robust function over a robust set and extend the optimality conditions of [3] and the algorithm of [4,5] to our case. The algorithm is shown to be effective.This research was supported by the National Science Foundation of China.     

Robust optimization with simulated annealing

Complex systems can be optimized to improve the performance with respect to desired functionalities. An optimized solution, however, can become suboptimal or even infeasible, when errors in implementation or input data are encountered. We report on a robust simulated annealing algorithm that does not require any knowledge of the problems structure. This is necessary in many engineering applications where solutions are often not explicitly known and have to be obtained by numerical simulations. While this nonconvex and global optimization method improves the performance as well as the robustness, it also warrants for a global optimum which is robust against data and implementation uncertainties. We demonstrate it on a polynomial optimization problem and on a high-dimensional and complex nanophotonic engineering problem and show significant improvements in efficiency as well as in actual optimality.     

Robust optimization using computer experiments

During metamodel-based optimization three types of implicit errors are typically made. The first error is the simulation-model error, which is defined by the difference between reality and the computer model. The second error is the metamodel error, which is defined by the difference between the computer model and the metamodel. The third is the implementation error. This paper presents new ideas on how to cope with these errors during optimization, in such a way that the final solution is robust with respect to these errors. We apply the robust counterpart theory of Ben-Tal and Nemirovsky to the most frequently used metamodels: linear regression and Kriging models. The methods proposed are applied to the design of two parts of the TV tube. The simulation-model errors receive little attention in the literature, while in practice these errors may have a significant impact due to propagation of such errors.     

Robust portfolio optimization with copulas

Conditional Value at Risk (CVaR) is widely used in portfolio optimization as a measure of risk. CVaR is clearly dependent on the underlying probability distribution of the portfolio. We show how copulas can be introduced to any problem that involves distributions and how they can provide solutions for the modeling of the portfolio. We use this to provide the copula formulation of the CVaR of a portfolio. Given the critical dependence of CVaR on the underlying distribution, we use a robust framework to extend our approach to Worst Case CVaR (WCVaR). WCVaR is achieved through the use of rival copulas. These rival copulas have the advantage of exploiting a variety of dependence structures, symmetric and not. We compare our model against two other models, Gaussian CVaR and Worst Case Markowitz. Our empirical analysis shows that WCVaR can asses the risk more adequately than the two competitive models during periods of crisis.     

Robust Design using classical optimization

The complexity of technical products increases significantly, due to an increasing number of interacting design variables of many components and subsystems. At the same time, the need for separated development processes is increasing due to specialization and outsourcing. Solution space methods are designed to solve this conflict. The requirements from an upper level, e.g. performance measures of the whole system, can be cascaded down to requirements on a lower level, e.g. performance measures of the subsystems or components, as it is done in the V-model approach. The method does not only take the numerous interactions into account but also guarantees the resulting intervals of different parameters to be independent of each other. Unfortunately, the computational cost of the state-of-the-art stochastic approach is high. The approach in this paper shows that the computational effort can be reduced considerably using a gradient based optimization approach for constraint problems. We demonstrate that the approach reduces the required number of function evaluations for a chassis design problem by a factor of 30, but more important, the CPU time for solving the problem by a factor of 20. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Robust ranking and portfolio optimization

The portfolio optimization problem has attracted researchers from many disciplines to resolve the issue of poor out-of-sample performance due to estimation errors in the expected returns. A practical method for portfolio construction is to use assets’ ordering information, expressed in the form of preferences over the stocks, instead of the exact expected returns. Due to the fact that the ranking itself is often described with uncertainty, we introduce a generic robust ranking model and apply it to portfolio optimization. In this problem, there are n objects whose ranking is in a discrete uncertainty set. We want to find a weight vector that maximizes some generic objective function for the worst realization of the ranking. This robust ranking problem is a mixed integer minimax problem and is very difficult to solve in general. To solve this robust ranking problem, we apply the constraint generation method, where constraints are efficiently generated by solving a network flow problem. For empirical tests, we use post-earnings-announcement drifts to obtain ranking uncertainty sets for the stocks in the DJIA index. We demonstrate that our robust portfolios produce smaller risk compared to their non-robust counterparts.     

Robust global optimization with polynomials

We consider the optimization problems max_z∈Ω min_x∈K p(z, x) and min_{x ∈ K} max_{z ∈ Ω} p(z, x) where the criterion p is a polynomial, linear in the variables z, the set Ω can be described by LMIs, and K is a basic closed semi-algebraic set. The first problem is a robust analogue of the generic SDP problem max_{z ∈ Ω} p(z), whereas the second problem is a robust analogue of the generic problem min_{x ∈ K} p(x) of minimizing a polynomial over a semi-algebraic set. We show that the optimal values of both robust optimization problems can be approximated as closely as desired, by solving a hierarchy of SDP relaxations. We also relate and compare the SDP relaxations associated with the max-min and the min-max robust optimization problems.     

Robust design optimization for egressing pedestrians in unknown environments

In this paper, we deal with a group variable in size of pedestrians moving in a unknown confined environment and searching for an exit. Pedestrian dynamics are simulated by means of a recently introduced microscopic (agent-based) model, characterized by an exploration phase and an egress phase. First, we study the model to reveal the role of its main parameters and its qualitative properties. Second, we tackle a robust optimization problem by means of the Particle Swarm Optimization method, aiming at reducing the time-to-target by adding in the walking area multiple obstacles optimally placed and shaped. Robustness is sought against the number of people in the group, which is an uncertain quantity described by a random variable with given probability density distribution.     

Robust topology optimization for multi-material structures under interval uncertainty

In this paper, we propose an efficient method to design robust multi-material structures under interval loading uncertainty. The objective of this study is to minimize the structural compliance of linear elastic structures. First, the loading uncertainty can be decomposed into two unit forces in the horizontal and vertical directions based on the orthogonal decomposition, which separates the uncertainty into the calculation coefficients of structural compliance that are not related to the finite element analysis. In this manner, the time-consuming procedure, namely, the nested double-loop optimization, can be avoided. Second, the uncertainty problem can be transformed into an augmented deterministic problem by means of uniform sampling, which exploits the coefficients related to interval variables. Finally, an efficient sensitivity analysis method is explicitly developed. Thus, the robust topology optimization (RTO) problem considering interval uncertainty can be solved by combining orthogonal decomposition with uniform sampling (ODUS). In order to eliminate the influence of numerical units when comparing the optimal results to deterministic and RTO solutions, the relative uncertainty related to interval objective function is employed to characterize the structural robustness. Several multi-material structure optimization cases are provided to demonstrate the feasibility and efficiency of the proposed method, where the magnitude uncertainty, directional uncertainty, and combined uncertainty are investigated.