Cutting-set methods for robust convex optimization with pessimizing oracles

A robust optimum solution is defined as one that is optimum and maintains its feasibility regardless of the values taken by uncertain parameters within a bounded interval. Solving for a robust optimum solution, however, can be computationally costly. To reduce the computational cost, a new robust optimization method, surrogate feasibility testing–cutting robust optimization (SFTC-RO), is proposed that solves two optimization subproblems. In the first subproblem, a scenario robust optimization problem is solved using a local optimization technique. In the second subproblem, a feasibility check and constraint cut of the feasible domain are performed via operations on a surrogate model of the constraints. The two subproblems are iteratively solved until convergence at a robust optimum solution. Results from numerical and engineering examples show that on average the proposed method can obtain robust optimal solutions at a lower computational cost and with better scalability than an existing method from the literature.

Variable fidelity robust optimization of pulsed laser orbital debris removal under epistemic uncertainty

Chapter 4 - Data-driven robust stochastic optimization for power systems operations

Robust optimization and stochastic programming approaches for medium-term production scheduling of a large-scale steelmaking continuous casting process under demand uncertainty

In order to improve the aerodynamic stability of a two-dimensional airfoil with deformable trailing edge during the change of external conditions,a robust optimization method considering the uncertainty was proposed. Based on class-shape function transformation( CST) method,a parametric model was established to represent the geometry of a two-dimensional airfoil with a deformable trailing edge. The differences between robust optimization method and deterministic optimization method were discussed. Taking into account the uncertain airfoil geometry and inlet Mach number,the robust optimization method was applied to maximize the mean value of the lift-to-drag ratio and minimize its standard deviation. The actuation power requirement of the optimized airfoil with a deformable trailing edge was calculated. The results show that the robust optimization method can help to improve the aerodynamic performance of morphing airfoil and reduce the sensitivity of this performance to inlet Mach number simultaneously and to reduce the actuation power requirement of the robust optimal airfoil.

Two-stage robust optimization for space heating loads of buildings in integrated community energy systems

This paper presents a new approach aims to solve robust multidisciplinary design optimization MDO problem called Improved Multi-objective Robust Collaborative Optimization . This method combines the Multi-objective Robust Collaborative Optimization method, the Worst Possible Point constraint cuts and the Genetic algorithm NSGA-II type as an optimizer to solve the robust optimization problem of complex structure named Y-stiffened panel under interval uncertainty . The proposed approach hierarchically decomposes the optimization problem into a structure level considered as an upper level in the Y-stiffened panel and a second level considered as a lower level of the studied panel. A robust multi-objective optimization problem intended to optimize the eigenfrequency, the global mass and the displacement at a fixed point of the Y-stiffened panel at the first level and each structure’s robust optimization problem allows optimizing its eigenfrequency and mass limited by their local constraint functions at the second one. Tor demonstrate our method, an engineering example of Y-stiffened panel is treated. A good performance of proposed method is proved by a comparison between obtained results and Non-Distributed Multi-objective Robust Optimization .

ABSTRACTThe rapid development of renewable energy has promoted the research of the integrated energy system. In particular, the joint optimal scheduling of renewable energy and traditional thermal power units is the key technology to solve the current renewable energy integration and the source network load balance. At present, the total installed capacity of cogeneration units is huge, which hinders the integration of renewable energy and the flexibility of the power grid regulation. Therefore, this paper proposes a robust optimal scheduling strategy for the plant‐level integrated energy system considering electricity‐heat balance. The wind power and solar energy are connected in parallel with the heating network of cogeneration units to decouple heat and power in a more economical and flexible way, while improving the integration of renewable energy. Based on the principle of Kang's robust optimization, robust optimization, a linear robust optimization probability density bias prediction optimization method for the plant‐level integrated energy system considering the electric‐heat balance is proposed in this paper. The dynamic nonlinear constraints in the output process of thermal and electrical loads are transformed into linear, which takes into account the conservatism of robust optimization and the economy of the objective function. The proposed optimization algorithm facilitates the integration of a high proportion of renewable energy into the power grid and is also applicable to other integrated energy systems with nonlinear constraint characteristics.

Robust optimization techniques attempt to find a solution that is both optimum and relatively insensitive to input uncertainty. In general, these techniques are computationally more expensive than their deterministic counterparts. In this article two new robust optimization methods are presented. The first method is called gradient-assisted robust optimization (GARO). In GARO, a robust optimization problem is first converted to a deterministic one by using a gradient-based approximation technique. After solving this deterministic problem, the solution robustness and the accuracy of the approximation are checked. If the accuracy meets a threshold, a robust optimum solution is found; otherwise, the approximation is adaptively modified until the threshold is met and a solution, if it exists, is obtained. The second method is a faster version of GARO called quasi-concave gradient-assisted robust optimization (QC-GARO). QC-GARO is for problems with quasi-concave objective and constraint functions. The difference between GARO and QC-GARO is in the way that they check the approximation accuracy. Both GARO and QC-GARO methods are applied to a set of six engineering design test problems and the results are compared with a few related previous methods. It was found that, compared to the methods considered, GARO could solve all test problems but with a higher computational effort compared to QC-GARO. However, QC-GARO was computationally much faster when it was able to solve the problems.

Multi-objective robust dynamic pricing and operation strategy optimization for integrated energy system based on stackelberg game

This paper is concerned with the problem of robust optimization method for coupling parametrization design of space observation system. The objective is to develop an robust optimization framework to design space observation system. An improved robust collaborative optimization (IRCO) method for complex system design using the hybrid optimization algorithm is developed, which associates genetic algorithm (GA) with tabu search (TS). The proposed method can avoid unsolvability or converging difficulty and handle uncertainty within and across disciplines more efficiently and reasonably. The results demonstrate the effectiveness and applicability of the proposed method.

This thesis explores possibilities and limitations of extending classical combinatorial optimization problems for network flows and network design. We propose new mathematical models for logistics networks that feature commodities with multidimensional properties, e.g. their mass and volume, to capture consolidation effects of commodities with complementing properties. We provide new theoretical insights and solution methods with immediate practical impact that we test on real-world instances from the automotive, chemical, and retail industry. The first model is for tactical transportation planning with temporal consolidation effects. We propose various heuristics and prove for our instances, that most of our solutions are within a single-digit percentage of the optimum. We also study problem variants where commodities are routed unsplittably and give hardness results for various special cases and a dynamic program that finds optimal forest solutions, which overestimate real costs. The second model is for strategic route planning under uncertainty. We provide for a robust optimization method that anticipates fluctuations of demands by minimizing worst-case costs over a restricted scenario set. We show that the adversary problem is NP-hard. To still find solutions with very good worst-case cost, we derive a carefully relaxed and simplified MILP, which solves well for large instances. It can be extended to include hub decisions leading to a robust M-median hub location problem. We find a price of robustness for our instances that is moderate for scenarios using average demand values as lower bounds. Trend based scenarios show a considerable tradeoff between historical average costs and worst case costs. Another robustness concept are incremental hub chains that provide solutions for every number of hubs to operate, such that they are robust under changes of this number. A comparison of incremental solutions with M-median solutions obtained with an LP-based search suggests that a price of being incremental is low for our instances. Finally, we investigate the problem of scheduling the maintenance of edges in a network. We focus on maintaining connectivity between two nodes over time. We show that the problem can be solved in polynomial time in arbitrary networks if preemption is allowed. If preemption is restricted to integral time points, the problem is NP-hard and for the non-preemptive case, we show strong non-approximability results.

This paper addresses the problem of robust optimization in large-scale networks of identical processors. General convex optimization problems are considered, where uncertain constraints are distributed to the processors in the network. The processors have to compute a maximizer of a linear objective over the robustly feasible set, defined as the intersection of all locally known feasible sets. We propose a novel asynchronous algorithm, based on outer-approximations of the robustly feasible set, to solve such problems. Each processor stores a small set of linear constraints that form an outer-approximation of the robustly feasible set. Based on its locally available information and the data exchange with neighboring processors, each processor repeatedly updates its local approximation. A computational study for robust linear programming illustrates that the completion time of the algorithm depends primarily on the diameter of the communication graph and is independent of the network size.

In engineering design, an optimized solution often turns out to be suboptimal when errors are encountered. Although the theory of robust convex optimization has taken significant strides over the past decade, all approaches fail if the underlying cost function is not explicitly given; it is even worse if the cost function is nonconvex. In this work, we present a robust optimization method that is suited for unconstrained problems with a nonconvex cost function as well as for problems based on simulations, such as large partial differential equations (PDE) solver, response surface, and Kriging metamodels. Moreover, this technique can be employed for most real-world problems because it operates directly on the response surface and does not assume any specific structure of the problem. We present this algorithm along with the application to an actual engineering problem in electromagnetic multiple scattering of aperiodically arranged dielectrics, relevant to nanophotonic design. The corresponding objective function is highly nonconvex and resides in a 100-dimensional design space. Starting from an “optimized” design, we report a robust solution with a significantly lower worst-case cost, while maintaining optimality. We further generalize this algorithm to address a nonconvex optimization problem under both implementation errors and parameter uncertainties.

Purpose: To develop the first 4D robust optimization (RO) method accounting for respiratory motion and evaluate its potential to improve plan robustness and optimality compared to 3D RO and PTV-based optimization. Methods: A set of 4D CT images are used to track respiratory motion and deformation of tumors and organs. For each of 10 respiration phases, dose distributions for nine different uncertainty scenarios including the nominal one, those incorporating ±5mm setup uncertainties long x, y and z directions and ±3.5% range uncertainties are calculated. All 90 dose distributions are simultaneously optimized to achieve full dose coverage of 10 CTVs and sparing of normal structures. ITV-based 3D RO and PTV-based optimization based on the average CT are also carried for the same patient using same dose volume constrains. After optimization, 4D robustness evaluation was performed for all resulting plans. The CTV coverage and the sparing of normal tissue in 10 phases are evaluated and compared among the three methods. The widths of DVH bands represent the robustness of dose distributions in the structures. Results: For one patient studied so far, the worst case CTV coverage by the prescription dose among all 90 scenarios is: 99% for 4D RO; 88.9% for 3D RO, and 85.2% for PTV based optimization. 4D RO also results in best robustness with the narrowest DVH’ bandwidths for the CTV. 4D and 3D RO have similar organ sparing while PTV based optimization results in worst organ sparing. Conclusion: 4D robust optimization which accounts for anatomy motion and deformation in the optimization process, significantly improves plan robustness and achieves higher quality treatment plans for lung cancer patients. The method is being evaluated for multiple patients with different tumor and motion characteristics.