Distributed robust optimization via Cutting-Plane Consensus

We consider a general multi-agent convex optimization problem where the agents are to collectively minimize a global objective function subject to a global inequality constraint, a global equality constraint, and a global constraint set. The objective function is defined by a sum of local objective functions, while the global constraint set is produced by the intersection of local constraint sets. In particular, we study two cases: one where the equality constraint is absent, and the other where the local constraint sets are identical. We devise two distributed primal-dual subgradient algorithms based on the characterization of the primal-dual optimal solutions as the saddle points of the Lagrangian and penalty functions. These algorithms can be implemented over networks with dynamically changing topologies but satisfying a standard connectivity property, and allow the agents to asymptotically agree on optimal solutions and optimal values of the optimization problem under the Slater's condition.

Optimization problems are used to model many real-life problems. Therefore, solving these problems is one of the most important goals of algorithm design. A general optimization problem can be defined by specifying a set of constraints that defines a subset in some underlying space (like the Euclidean space) called the feasible subset and an objective function that we are trying to maximize or minimize, as the case may be, over the feasible set. The difficulty of solving such problems typically depends on how ‘complex’ the feasible set and the objective function are. For example, a very important class of optimization problems is linear programming. Here the feasible subset is specified by a set of linear inequalities (in the Euclidean space); the objective function is also linear. A more general class of optimization problems is convex programming, where the feasible set is a convex subset of a Euclidean space and the objective function is also convex. Convex programs (and hence, linear programs) have a nice property that any local optimum is also a global optimum for the objective function. There are a variety of techniques for solving such problems – all of them try to approach a local optimum (which we know would be a global optimum as well). These notions are discussed in greater detail in a later section in this chapter. The more general problem, the so-called non-convex programs, where the objective function and the feasible subset could be arbitrary can be very challenging to solve. In particular, discrete optimization problems, where the feasible subset could be a (large) discrete subset of points falls under this category.

We consider a general worst-case robust convex optimization problem, with arbitrary dependence on the uncertain parameters, which are assumed to lie in some given set of possible values. We describe a general method for solving such a problem, which alternates between optimization and worst-case analysis. With exact worst-case analysis, the method is shown to converge to a robust optimal point. With approximate worst-case analysis, which is the best we can do in many practical cases, the method seems to work very well in practice, subject to the errors in our worst-case analysis. We give variations on the basic method that can give enhanced convergence, reduce data storage, or improve other algorithm properties. Numerical simulations suggest that the method finds a quite robust solution within a few tens of steps; using warm-start techniques in the optimization steps reduces the overall effort to a modest multiple of solving a nominal problem, ignoring the parameter variation. The method is illustrated with several application examples.

Due to the generation uncertainty of photovoltaic (PV) power generation, it has been posing great challenges and difficulties in maintaining the stability, security, and reliability of PV-storage systems (one kind of microgrid). To overcome these challenges and difficulties, this paper is concerned with secondary control and robust energy management for PVs in a grid-connected microgrid (MG) considering uncertainty. In our designs, to maintain the stable operation of PVs in MG, a novel secondary control method combining an event-triggered finite time sliding mode controller (FTSMC) and consensus controllers is proposed. Furthermore, a robust optimization framework is established to minimize the total cost of grid-connected MG involving the operation cost of multi-battery Energy Storage Systems (BESSes) and the electricity purchased from the main grid. To eliminate the effects of PV uncertainty, the optimization problem with uncertain constraints is converted into a new optimization problem with only deterministic constraints by using the box theory to represent the PV outputs. In other words, the robust optimization strategy makes uncertain boundaries easier to be represented by setting all uncertain parameters into an uncertain domain involving all typical extreme cases. Then, a particle swarm optimization (PSO) method is employed to solve the newly converted optimization problem. Finally, the experimental results validate the effectiveness of the proposed integrated framework.

In this paper, we consider adjustable robust versions of convex optimization problems with uncertain constraints and objectives and show that under fairly general assumptions, a static robust solution provides a good approximation for these adjustable robust problems. An adjustable robust optimization problem is usually intractable since it requires to compute a solution for all possible realizations of uncertain parameters, while an optimal static solution can be computed efficiently in most cases if the corresponding deterministic problem is tractable. The performance of the optimal static robust solution is related to a fundamental geometric property, namely, the symmetry of the uncertainty set. Our work allows for the constraint and objective function coefficients to be uncertain and for the constraints and objective functions to be convex, thereby providing significant extensions of the results in Bertsimas and Goyal (Math Oper Res 35:284–305, 2010) and Bertsimas et al. (Math Oper Res 36: 24–54, 2011b) where only linear objective and linear constraints were considered. The models in this paper encompass a wide variety of problems in revenue management, resource allocation under uncertainty, scheduling problems with uncertain processing times, semidefinite optimization among many others. To the best of our knowledge, these are the first approximation bounds for adjustable robust convex optimization problems in such generality.

Fast robust optimization of ORC based on an artificial neural network for waste heat recovery

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 .

Inexact alternating direction multiplier methods (ADMMs) are developed for solving general separable convex optimization problems with a linear constraint and with an objective that is the sum of smooth and nonsmooth terms. The approach involves linearized subproblems, a back substitution step, and either gradient or accelerated gradient techniques. Global convergence is established. The methods are particularly useful when the ADMM subproblems do not have closed form solution or when the solution of the subproblems is expensive. Numerical experiments based on image reconstruction problems show the effectiveness of the proposed methods.

Recently the authors have proposed a homogeneous and self-dual algorithm for solving the monotone complementarity problem (MCP) [5]. The algorithm is a single phase interior-point type method; nevertheless, it yields either an approximate optimal solution or detects a possible infeasibility of the problem. In this paper we specialize the algorithm to the solution of general smooth convex optimization problems, which also possess nonlinear inequality constraints and free variables. We discuss an implementation of the algorithm for large-scale sparse convex optimization. Moreover, we present computational results for solving quadratically constrained quadratic programming and geometric programming problems, where some of the problems contain more than 100,000 constraints and variables. The results indicate that the proposed algorithm is also practically efficient.

A subgradient method is presented for solving general convex optimization problems, the main requirement being that a strictly feasible point is known. A feasible sequence of iterates is generated, which converges to within user-specified error of optimality. Feasibility is maintained with a line search at each iteration, avoiding the need for orthogonal projections onto the feasible region (an operation that limits practicality of traditional subgradient methods). Lipschitz continuity is not required, yet the algorithm is shown to possess a convergence rate analogous to rates for traditional methods, albeit with error measured relatively, whereas traditionally error has been absolute. The algorithm is derived using an elementary framework that can be utilized to design other such algorithms.

A parallelized DPPQN based Expert System method and implementation

Surrogate duality for robust optimization

One critical issue for manufacturing 3D MCM packages is the warpage induced during the process. The effects of several important parameters, including processing conditions, package geometry and materials, are specifically studied on the occurrence of warpage and optimized to reduce the warpage in many previous literatures. However, the design parameter values are not always precisely known but can contain a level of uncertainty to some extent due to, for example manufacturing tolerance. In some case, the design parameters variations can not be neglected. In this paper, the effect of structure geometry tolerance and material property tolerance on package warpage robustness was took into account and analyzed. The robust optimization mathematic model of the 3D packages, considering the uncertainty of the design parameters, was built. And the robust optimization method based on the Monte-carlo simulations and Genetic Algorithm was present to solve the optimization problem above. Firstly, the parametric design finite element model of the specific 3D MCM package was built. Secondly, the statistic distribution of the warpage induced by the design parameters variations was obtained based on the Monte-carlo simulations. Finally, the GA was implemented to optimize the mean value and standard deviation of the warpage. The resulting robust optimization design shows that the robust optimization design is insensitive to variations on design variables within a feasible region.

Multilevel optimization problems deals with mathematical programming problems whose feasible set is implicitly determined by a sequence of nested optimization problems. These kind of problems are common in different applications where there is a hierarchy of decision makers exists. Solving such problems has been a challenge especially when they are non linear and non convex. In this paper we introduce a new algorithm, inspired by natural adaptation, using (1+1)-evolutionary strategy iteratively. Suppose there are k level optimization problem. First, the leader’s level will be solved alone for all the variables under all the constraint set. Then that solution will adapt itself according to the objective function in each level going through all the levels down. When a particular level’s optimization problem is solved the solution will be adapted the level’s variable while the other variables remain being a fixed parameter. This updating process of the solution continues until a stopping criterion is met. Bilevel and trilevel optimization problems are used to show how the algorithm works. From the simulation result on the two problems, it is shown that it is promising to uses the proposed metaheuristic algorithm in solving multilevel optimization problems.KeywordsMultilevel optimization(1+1)-Evolutionary strategymetaheuristic algorithmsNatural adaptation

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