11,702 research outputs found
Integer Programming Approaches for Some Non-convex and Stochastic Optimization Problems
In this dissertation we study several non-convex and stochastic optimization problems. The common theme is the use of mixed-integer programming (MIP) techniques including valid inequalities and reformulation to solve these problems.
We first study a strategic capacity planning model which captures the trade-off between the incentive to delay capacity installation to wait for improved technology and the need for some capacity to be installed to meet current demands. This problem is naturally formulated as a MIP with a bilinear objective. We develop several linear MIP formulations, along with classes of strong valid inequalities. We also present a specialized branch-and-cut algorithm to solve a compact concave formulation. Computational results indicate that these formulations can be used to solve large-scale instances.
We next study methods for optimization with joint probabilistic constraints. These problems are challenging because evaluating solution feasibility requires multidimensional integration and the feasible region is not convex. We propose and analyze a Monte Carlo sampling scheme to simplify the probabilistic structure of such problems. Computational tests of the approach indicate that it can yield good feasible solutions and reasonable bounds on their quality. Next, we study a MIP formulation of the non-convex sample approximation problem. We obtain two strengthened formulations. As a byproduct of this analysis, we obtain new results for the previously studied mixing set, subject to an additional knapsack inequality. Computational results indicate that large-scale instances can be solved using the strengthened formulations.
Finally, we study optimization problems with stochastic dominance constraints. A stochastic dominance constraint states that a random outcome which depends on the decision variables should stochastically dominate a given random variable. We present new formulations for both first and second order stochastic dominance which are significantly more compact than existing formulations. Computational tests illustrate the benefits of the new formulations.Ph.D.Committee Co-Chair: Ahmed, Shabbir; Committee Co-Chair: Nemhauser, George; Committee Member: Cook, Bill; Committee Member: Gu, Zonghao; Committee Member: Parker, R. Gar
A scenario approach for non-convex control design
Randomized optimization is an established tool for control design with
modulated robustness. While for uncertain convex programs there exist
randomized approaches with efficient sampling, this is not the case for
non-convex problems. Approaches based on statistical learning theory are
applicable to non-convex problems, but they usually are conservative in terms
of performance and require high sample complexity to achieve the desired
probabilistic guarantees. In this paper, we derive a novel scenario approach
for a wide class of random non-convex programs, with a sample complexity
similar to that of uncertain convex programs and with probabilistic guarantees
that hold not only for the optimal solution of the scenario program, but for
all feasible solutions inside a set of a-priori chosen complexity. We also
address measure-theoretic issues for uncertain convex and non-convex programs.
Among the family of non-convex control- design problems that can be addressed
via randomization, we apply our scenario approach to randomized Model
Predictive Control for chance-constrained nonlinear control-affine systems.Comment: Submitted to IEEE Transactions on Automatic Contro
Mixed-Integer Convex Nonlinear Optimization with Gradient-Boosted Trees Embedded
Decision trees usefully represent sparse, high dimensional and noisy data.
Having learned a function from this data, we may want to thereafter integrate
the function into a larger decision-making problem, e.g., for picking the best
chemical process catalyst. We study a large-scale, industrially-relevant
mixed-integer nonlinear nonconvex optimization problem involving both
gradient-boosted trees and penalty functions mitigating risk. This
mixed-integer optimization problem with convex penalty terms broadly applies to
optimizing pre-trained regression tree models. Decision makers may wish to
optimize discrete models to repurpose legacy predictive models, or they may
wish to optimize a discrete model that particularly well-represents a data set.
We develop several heuristic methods to find feasible solutions, and an exact,
branch-and-bound algorithm leveraging structural properties of the
gradient-boosted trees and penalty functions. We computationally test our
methods on concrete mixture design instance and a chemical catalysis industrial
instance
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