On the mixing set with a knapsack constraint

We study a substructure appearing in mixed-integer programming reformulations of chance-constrained programs with stochastic right-hand-sides over a finite discrete distribution, which we call the mixing set with a knapsack constraint. Recently, Luedtke et al. (Math. Program. 122(2):247–272, 2010) and Küçükyavuz (Math Program 132(1):31–56, 2012) studied valid inequalities for such sets. However, most of their results were focused on the equal probabilities case (when the knapsack constraint reduces to a cardinality constraint). In this paper, we focus on the general probabilities case (general knapsack constraint). We characterize the valid inequalities that do not come from the knapsack polytope and use this characterization to generalize the results previously derived for the equal probabilities case. Our results allow for a deep understanding of the relationship that the set under consideration has with the knapsack polytope. Moreover, they allow us to establish benchmarks that can be used to identify when a relaxation will be useful for the considered types of reformulations of chance-constrained programs

An augmented lagrangian decomposition method for chance-constrained optimization problems

Joint chance-constrained optimization problems under discrete distributions arise frequently in financial management and business operations. These problems can be reformulated as mixed-integer programs. The size of reformulated integer programs is usually very large even though the original problem is of medium size. This paper studies an augmented Lagrangian decomposition method for finding high-quality feasible solutions of complex optimization problems, including nonconvex chance-constrained problems. Different from the current augmented Lagrangian approaches, the proposed method allows randomness to appear in both the left-hand-side matrix and the right-hand-side vector of the chance constraint. In addition, the proposed method only requires solving a convex subproblem and a 0-1 knapsack subproblem at each iteration. Based on the special structure of the chance constraint, the 0-1 knapsack problem can be computed in quasi-linear time, which keeps the computation for discrete optimization subproblems at a relatively low level. The convergence of the method to a first-order stationary point is established under certain mild conditions. Numerical results are presented in comparison with a set of existing methods in the literature for various real-world models. It is observed that the proposed method compares favorably in terms of the quality of the best feasible solution obtained within a certain time for large-size problems, particularly when the objective function of the problem is nonconvex or the left-hand-side matrix of the constraints is random

Electrical Infrastructure Adaptation for a Changing Climate

In recent years, global climate change has become a major factor in long-term electrical infrastructure planning in coastal areas. Over time, accelerated sea level rise and fiercer, more frequent storm surges caused by the changing climate have imposed increasing risks to the security and reliability of coastal electrical infrastructure systems. It is important to ensure that infrastructure system planning adapts to such risks to produce systems with strong resilience. This dissertation proposes a decision framework for long-term, resilient electrical infrastructure adaptation planning for a future with the uncertain sea level rise and storm surges in a changing climate. As uncertainty is unavoidable in real-world decision making, stochastic optimization plays an essential role in making robust decisions with respect to global climate change. The core of the proposed decision framework is a stochastic optimization model with the primary goal being to ensure operational feasibility once uncertain futures are revealed. The proposed stochastic model produces long-term climate adaptations that are subject to both the exogenous uncertainty of climate change as well as the endogenous physical restrictions of electrical infrastructure. Complex, state-of-the-art simulation models under climate change are utilized to represent exogenous uncertainty in the decision-making process. In practice, deterministic methods such as scenario-based analyses and/or geometric-information-system-based heuristics are widely used for real-world adaptation planning. Numerical experiments and sensitivity analyses are conducted to compare the proposed framework with various deterministic methods. Our experimental results demonstrate that resilient, long-term adaptations can be obtained using the proposed stochastic optimization model. In further developing the decision framework, we address a class of stochastic optimization models where operational feasibility is ensured for only a percentage of all possible uncertainty realizations through joint chance-constraints. It is important to identify the significant scalability limitations often associated with commercial optimization tools for solving this class of challenging stochastic optimization problems. We propose a novel configuration generation algorithm which leverages metaheuristics to find high-quality solutions quickly and generic relaxations to provide solution quality guarantees. A key advantage of the proposed method over previous work is that the joint chance-constrained stochastic optimization problem can contain multivariate distributions, discrete variables, and nonconvex constraints. The effectiveness of the proposed algorithm is demonstrated on two applications, including the climate adaptation problem, where it significantly outperforms commercial optimization tools. Furthermore, the need to address the feasibility of a realistic electrical infrastructure system under impacts is recognized for the proposed decision framework. This requires dedicated attention to addressing nonlinear, nonconvex optimization problem feasibility, which can be a challenging problem that requires an expansive exploitation of the solution space. We propose a global algorithm for the feasibility problem\u27s counterpart: proving problem infeasibility. The proposed algorithm adaptively discretizes variable domains to tighten the relaxed problem for proving infeasibility. The convergence of the algorithm is demonstrated as the algorithm either finds a feasible solution or terminates with the problem being proven infeasible. The efficiency of this algorithm is demonstrated through experiments comparing two state-of-the-art global solvers, as well as a recently proposed global algorithm, to our proposed method

New Solution Methods for Joint Chance-Constrained Stochastic Programs with Random Left-Hand Sides

We consider joint chance-constrained programs with random lefthand sides. The motivation of this project is that this class of problem has many important applications, but there are few existing solution methods. For the most part, we deal with the subclass of problems for which the underlying parameter distributions are discrete. This assumption allows the original problem to be formulated as a deterministic equivalent mixed-integer program. We rst approach the problem as a mixed-integer program and derive a class of optimality cuts based on irreducibly infeasible subsets of the constraints of the scenarios of the problem. The IIS cuts can be computed effciently by means of a linear program. We give a method for improving the upper bound of the problem when no IIS cut can be identifi ed. We also give an implementation of an algorithm incorporating these ideas and finish with some computational results. We present a tabu search metaheuristic for fi nding good feasible solutions to the mixed-integer formulation of the problem. Our heuristic works by de ning a sufficient set of scenarios with the characteristic that all other scenarios do not have to be considered when generating upper bounds. We then use tabu search on the one-opt neighborhood of the problem. We give computational results that show our metaheuristic outperforming the state-of-the-art industrial solvers. We then show how to reformulate the problem so that the chance-constraints are monotonic functions. We then derive a convergent global branch-and-bound algorithm using the principles of monotonic optimization. We give a finitely convergent modi cation of the algorithm. Finally, we give a discussion on why this algorithm is computationally ine ffective. The last section of this dissertation details an application of joint chance-constrained stochastic programs to a vaccination allocation problem. We show why it is necessary to formulate the problem with random parameters and also why chance-constraints are a good framework for de fining an optimal policy. We give an example of the problem formulated as a chance constraint and a short numerical example to illustrate the concepts