Application of gradient based algorithms for robust shape optimization of a tensile test specimen

In case of technical applications such as automotive or aeronautic part optimization, uncertainties in design variables and other parameters have to be taken into account. Especially, scattering of material properties and mechanical loads have large influences on the component behavior. Due to the difficulty in calculating of sensitivities, usually stochastic algorithms are used for robust and reliability optimization. That entails a large number of function calls and long computation time. Therefore, the stochastic responses with expectation value and variances will be approximated by response surfaces in this contribution. Based on the practical example of a tensile test specimen a deterministic optimization and optimization under uncertainties will be done. The results of the different optimizations will be compared and the influence of the uncertainties will be shown and discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A robust robust optimization result

We study the loss in objective value when an inaccurate objective is optimized instead of the true one, and show that “on average” this loss is very small, for an arbitrary compact feasible region.     

Data-driven robust optimization

Mathematical Programming - The last decade witnessed an explosion in the availability of data for operations research applications. Motivated by this growing availability, we propose a novel schema...     

Optimization for automated assembly of puzzles

The puzzle-assembly problem has many application areas such as restoration and reconstruction of archeological findings, repairing of broken objects, solving jigsaw type puzzles, molecular docking problem, etc. The puzzle pieces usually include not only geometrical shape information but also visual information such as texture, color, and continuity of lines. This paper presents a new approach to the puzzle-assembly problem that is based on using textural features and geometrical constraints. The texture of a band outside the border of pieces is predicted by inpainting and texture synthesis methods. Feature values are derived from these original and predicted images of pieces. An affinity measure of corresponding pieces is defined and alignment of the puzzle pieces is formulated as an optimization problem where the optimum assembly of the pieces is achieved by maximizing the total affinity measure. A Fast Fourier Transform based image registration technique is used to speed up the alignment of the pieces. Experimental results are presented on real and artificial data sets.     

A dynamic programming approach to adjustable robust optimization

In this paper we consider the adjustable robust approach to multistage optimization, for which we derive dynamic programming equations. We also discuss this from the point of view of risk averse stochastic programming. We consider as an example a robust formulation of the classical inventory model and show that, like for the risk neutral case, a basestock policy is optimal.     

The dynamic programming approach to multi-model robust optimization

The aim of this paper is to extend the dynamic programming (DP) approach to multi-model optimal control problems (OCPs). We deal with robust optimization of multi-model control systems and are particularly interested in the Hamilton-Jacobi-Bellman (HJB) equation for the above class of problems. In this paper, we study a variant of the HJB for multi-model OCPs and examine the natural relationship between the Bellman DP techniques and the Robust Maximum Principle (MP). Moreover, we describe how to carry out the practical calculations in the context of multi-model LQ-problems and derive the associated Riccati-type equation.     

Efficient robust optimization for robust control with constraints

This paper proposes an efficient computational technique for the optimal control of linear discrete-time systems subject to bounded disturbances with mixed linear constraints on the states and inputs. The problem of computing an optimal state feedback control policy, given the current state, is non-convex. A recent breakthrough has been the application of robust optimization techniques to reparameterize this problem as a convex program. While the reparameterized problem is theoretically tractable, the number of variables is quadratic in the number of stages or horizon length N and has no apparent exploitable structure, leading to computational time of per iteration of an interior-point method. We focus on the case when the disturbance set is ∞-norm bounded or the linear map of a hypercube, and the cost function involves the minimization of a quadratic cost. Here we make use of state variables to regain a sparse problem structure that is related to the structure of the original problem, that is, the policy optimization problem may be decomposed into a set of coupled finite horizon control problems. This decomposition can then be formulated as a highly structured quadratic program, solvable by primal-dual interior-point methods in which each iteration requires time. This cubic iteration time can be guaranteed using a Riccati-based block factorization technique, which is standard in discrete-time optimal control. Numerical results are presented, using a standard sparse primal-dual interior point solver, that illustrate the efficiency of this approach.     

A robust optimization approach to wine grape harvesting scheduling

Optimization models are increasingly being used in agricultural planning. However, the inherent uncertainties present in agriculture make it difficult. In recent years, robust optimization has emerged as a methodology that allows dealing with uncertainty in optimization models, even when probabilistic knowledge of the phenomenon is incomplete. In this paper, we consider a wine grape harvesting scheduling optimization problem subject to several uncertainties, such as the actual productivity that can be achieved when harvesting. We study how effective robust optimization is solving this problem in practice. We develop alternative robust models and show results for some test problems obtained from actual wine industry problems.     

On scenario aggregation to approximate robust combinatorial optimization problems

As most robust combinatorial min–max and min–max regret problems with discrete uncertainty sets are NP-hard, research in approximation algorithm and approximability bounds has been a fruitful area of recent work. A simple and well-known approximation algorithm is the midpoint method, where one takes the average over all scenarios, and solves a problem of nominal type. Despite its simplicity, this method still gives the best-known bound on a wide range of problems, such as robust shortest path or robust assignment problems. In this paper, we present a simple extension of the midpoint method based on scenario aggregation, which improves the current best K-approximation result to an \((\varepsilon K)\)-approximation for any desired \(\varepsilon > 0\). Our method can be applied to min–max as well as min–max regret problems.     

A decision-theoretic approach to robust optimization in multivalued graphs

This paper is devoted to the search of robust solutions in finite graphs when costs depend on scenarios. We first point out similarities between robust optimization and multiobjective optimization. Then, we present axiomatic requirements for preference compatibility with the intuitive idea of robustness in a multiple scenarios decision context. This leads us to propose the Lorenz dominance rule as a basis for robustness analysis. Then, after presenting complexity results about the determination of Lorenz optima, we show how the search can be embedded in algorithms designed to enumerate k best solutions. Then, we apply it in order to enumerate Lorenz optimal spanning trees and paths. We investigate possible refinements of Lorenz dominance and we propose an axiomatic justification of OWA operators in this context. Finally, the results of numerical experiments on randomly generated graphs are provided. They show the numerical efficiency of the suggested approach.     

Applications of optimization methods to robust stability of linear systems

We study the robust stability problem for a family of polynomials. We allow for all the coefficients of the polynomials to be affinely perturbed, where the size of the perturbation is measured by an arbitrary convex function. We apply optimization techniques, and in particular convex duality methods, to derive simple formulas for the stability radius, to find a minimal perturbation which destroys stability, and to obtain necessary and sufficient conditions for robust stability. Our framework is general enough to cover many applications. As special cases, we obtain many results recently reported in the literature.The work of the first author was partially supported by AFOSR Grant 91-008 and NSF Grant DMS-92-01297.     

鲁棒优化中的Pareto有效性

对于一般的不确定优化问题, 研究了鲁棒解的~Pareto 有效性. 首先, 证明了Pareto 鲁棒解集即是鲁棒解集的Pareto 有效集, 因此求Pareto 鲁棒解等价于求鲁棒解集的Pareto 有效元. 其次, 基于推广的epsilon-约束方法, 得到了Pareto 鲁棒解的生成方法.     

Min-max and robust polynomial optimization

We consider the robust (or min-max) optimization problem

$J^*:=\max_{\mathbf{y}\in{\Omega}}\min_{\mathbf{x}}\{f(\mathbf{x},\mathbf{y}): (\mathbf{x},\mathbf{y})\in\mathbf{\Delta}\}$

where f is a polynomial and \({\mathbf{\Delta}\subset\mathbb{R}^n\times\mathbb{R}^p}\) as well as \({{\Omega}\subset\mathbb{R}^p}\) are compact basic semi-algebraic sets. We first provide a sequence of polynomial lower approximations \({(J_i)\subset\mathbb{R}[\mathbf{y}]}\) of the optimal value function \({J(\mathbf{y}):=\min_\mathbf{x}\{f(\mathbf{x},\mathbf{y}): (\mathbf{x},\mathbf{y})\in \mathbf{\Delta}\}}\). The polynomial \({J_i\in\mathbb{R}[\mathbf{y}]}\) is obtained from an optimal (or nearly optimal) solution of a semidefinite program, the ith in the “joint + marginal” hierarchy of semidefinite relaxations associated with the parametric optimization problem \({\mathbf{y}\mapsto J(\mathbf{y})}\), recently proposed in Lasserre (SIAM J Optim 20, 1995-2022, 2010). Then for fixed i, we consider the polynomial optimization problem \({J^*_i:=\max\nolimits_{\mathbf{y}}\{J_i(\mathbf{y}):\mathbf{y}\in{\Omega}\}}\) and prove that \({\hat{J}^*_i(:=\displaystyle\max\nolimits_{\ell=1,\ldots,i}J^*_\ell)}\) converges to J* as i → ∞. Finally, for fixed ? ≤ i, each \({J^*_\ell}\) (and hence \({\hat{J}^*_i}\)) can be approximated by solving a hierarchy of semidefinite relaxations as already described in Lasserre (SIAM J Optim 11, 796–817, 2001; Moments, Positive Polynomials and Their Applications. Imperial College Press, London 2009).

     

Surrogate duality for robust optimization

Robust optimization problems, which have uncertain data, are considered. We prove surrogate duality theorems for robust quasiconvex optimization problems and surrogate min–max duality theorems for robust convex optimization problems. We give necessary and sufficient constraint qualifications for surrogate duality and surrogate min–max duality, and show some examples at which such duality results are used effectively. Moreover, we obtain a surrogate duality theorem and a surrogate min–max duality theorem for semi-definite optimization problems in the face of data uncertainty.     

On robust optimization of two-stage systems

Robust-optimization models belong to a special class of stochastic programs, where the traditional expected cost minimization objective is replaced by one that explicitly addresses cost variability. This paper explores robust optimization in the context of two-stage planning systems. We show that, under arbitrary measures for variability, the robust optimization approach might lead to suboptimal solutions to the second-stage planning problem. As a result, the variability of the second-stage costs may be underestimated, thereby defeating the intended purpose of the model. We propose sufficient conditions on the variability measure to remedy this problem. Under the proposed conditions, a robust optimization model can be efficiently solved using a variant of the L-shaped decomposition algorithm for traditional stochastic linear programs. We apply the proposed framework to standard stochastic-programming test problems and to an application that arises in auctioning excess electric power. Mathematics Subject Classification (1991):90C15, 90C33, 90B50, 90A09, 90A43Supported in part by NSF Grants DMI-0099726 and DMI-0133943     

Decomposition for adjustable robust linear optimization subject to uncertainty polytope

We present in this paper a general decomposition framework to solve exactly adjustable robust linear optimization problems subject to polytope uncertainty. Our approach is based on replacing the polytope by the set of its extreme points and generating the extreme points on the fly within row generation or column-and-row generation algorithms. The novelty of our approach lies in formulating the separation problem as a feasibility problem instead of a max–min problem as done in recent works. Applying the Farkas lemma, we can reformulate the separation problem as a bilinear program, which is then linearized to obtained a mixed-integer linear programming formulation. We compare the two algorithms on a robust telecommunications network design under demand uncertainty and budgeted uncertainty polytope. Our results show that the relative performance of the algorithms depend on whether the budget is integer or fractional.     

鲁棒信用风险优化的线性锥优化模型

考虑了具有强健性的信用风险优化问题. 根据最差条件在值风险度量信用风险的方法,建立了信用风险优化问题的模型. 由于信用风险的损失分布存在不确定性,考虑了两类不确定性区间,即箱子型区间和椭球型区间. 把具有强健性的信用风险优化问题分别转化成线性规划问题和二阶锥规划问题. 最后,通过一个信用风险问题的例子来说明此模型的有效性.     

The application of automated reasoning to formal models of combinatorial optimization

Many formalisms have been proposed over the years to capture combinatorial optimization algorithms such as dynamic programming, branch and bound, and greedy. In 1989 Helman [9] presented a common formalism that captures dynamic programming and branch and bound type algorithms. The formalism was later extended to include greedy algorithms. In this paper, we describe the application of automated reasoning techniques to the domain of our model, in particular considering some representational issues and demonstrating that proofs about the model can be obtained by an automated reasoning program. The long-term objective of this research is to develop a methodology for using automated reasoning to establish new results within the theory, including the derivation of new lower bounds and the discovery (and verification) of new combinatorial search strategies.     

Semiconductor lot allocation using robust optimization

In this work, the problem of allocating a set of production lots to satisfy customer orders is considered. This research is of relevance to lot-to-order matching problems in semiconductor supply chain settings. We consider that lot-splitting is not allowed during the allocation process due to standard practices. Furthermore, lot-sizes are regarded as uncertain planning data when making the allocation decisions due to potential yield loss. In order to minimize the total penalties of demand un-fulfillment and over-fulfillment, a robust mixed-integer optimization approach is adopted to model is proposed the problem of allocating a set of work-in-process lots to customer orders, where lot-sizes are modeled using ellipsoidal uncertainty sets. To solve the optimization problem efficiently we apply the techniques of branch-and-price and Benders decomposition. The advantages of our model are that it can represent uncertainty in a straightforward manner with little distributional assumptions, and it can produce solutions that effectively hedge against the uncertainty in the lot-sizes using very reasonable amounts of computational effort.