an efficient hidden variable approach to minimal case camera motion estimation in this paper we present an efficient new approach for solving two view minimal case problems in camera motion estimation most notably the so called five point relative orientation problem and the six point focal length problem our approach is based on the hidden variable technique used in solving multivariate polynomial systems the resulting algorithm is conceptually simple which involves a relaxation which replaces monomials in all but one of the variables to reduce the problem to the solution of sets of linear equations as well as solving a polynomial eigenvalue problem polyeig to efficiently find the polynomial eigenvalues we make novel use of several numeric techniques which include quotient free gaussian elimination levinson durbin iteration and also a dedicated root polishing procedure we have tested the approach on different minimal cases and extensions with satisfactory results obtained both the executables and source codes of the proposed algorithms are made freely downloadable

min max and min max regret versions of combinatorial optimization problems a survey abstract min max and min max regret criteria are commonly used to define robust solutions after motivating the use of these criteria we present general results then we survey complexity results for the min max and min max regret versions of some combinatorial optimization problems shortest path spanning tree assignment min cut min s t cut knapsack since most of these problems are np hard we also investigate the approximability of these problems furthermore we present algorithms to solve these problems to optimality

a minmax regret approach to the critical path method with task interval times abstract the execution of a given project with a number of interrelated tasks due to precedence constraints represents a challenge when one must to control the available resources and the compromised due dates in this paper we analyse this problem under uncertain individual task completing times specifically we will assume that a given range for the admissible values of each individual completing time is available taking into account that the precedence relations between tasks must be preserved each realization of the admissible execution times for the set of tasks will define a new scenario determining the ending time for the project and the subset of critical tasks the minmax regret criterion will be used in order to obtain a robust approximation to the critical set of tasks determining the overall execution time for the project

minmax regret location allocation problem on a network under uncertainty abstract we consider a robust location allocation problem with uncertainty in demand coefficients specifically for each demand point only an interval estimate of its demand is known and we consider the problem of determining where to locate a new service when a given fraction of these demand points must be served by the utility the optimal solution of this problem is determined by the minimax regret location i e the point that minimizes the worst case loss in the objective function that may occur because a decision is made without knowing which state of nature will take place for the case where the demand points are vertices of a network we show that the robust location allocation problem can be solved in o min p n p n 3 m time where n is the number of demand points p p n is the fixed number of demand points that must be served by the new service and m is the number of edges of the network

two stage robust network flow and design under demand uncertainty we describe a two stage robust optimization approach for solving network flow and design problems with uncertain demand in two stage network optimization one defers a subset of the flow decisions until after the realization of the uncertain demand availability of such a recourse action allows one to come up with less conservative solutions compared to single stage optimization however this advantage often comes at a price two stage optimization is in general significantly harder than single stage optimization for network flow and design under demand uncertainty we give a characterization of the first stage robust decisions with an exponential number of constraints and prove that the corresponding separation problem is np hard even for a network flow problem on a bipartite graph we show however that if the second stage network topology is totally ordered or an arborescence then the separation problem is tractable unlike single stage robust optimization under demand uncertainty two stage robust optimization allows one to control conservatism of the solutions by means of an allowed budget for demand uncertainty using a budget of uncertainty we provide an upper bound on the probability of infeasibility of a robust solution for a random demand vector we generalize the approach to multicommodity network flow and design and give applications to lot sizing and location transportation problems by projecting out second stage flow variables we define an upper bounding problem for the two stage min max min optimization problem finally we present computational results comparing the proposed two stage robust optimization approach with single stage robust optimization as well as scenario based two stage stochastic optimization

variational recursive joint estimation of dense scene structure and camera motion from monocular high speed traffic sequences we present an approach to jointly estimating camera motion and dense structure of a static scene in terms of depth maps from monocular image sequences in driver assistance scenarios at each instant of time only two consecutive frames are processed as input data of a joint estimator that fully exploits second order information of the corresponding optimization problem and effectively copes with the non convexity due to both the imaging geometry and the manifold of motion parameters additionally carefully designed gaussian approximations enable probabilistic inference based on locally varying confidence and globally varying sensitivity due to the epipolar geometry with respect to the high dimensional depth map estimation embedding the resulting joint estimator in an online recursive framework achieves a pronounced spatio temporal filtering effect and robustness we evaluate hundreds of images taken from a car moving at speed up to 100 km h and being part of a publicly available benchmark data set the results compare favorably with two alternative settings stereo based scene reconstruction and camera motion estimation in batch mode using multiple frames they however require a calibrated camera pair or storage for more than two frames which is less attractive from a technical viewpoint than the proposed monocular and recursive approach in addition to real data a synthetic sequence is considered which provides reliable ground truth

on combinatorial optimization problems on matroids with uncertain weights in this paper the combinatorial optimization problem on weighted matroid is considered it is assumed that the weights in the problem are ill known and they are modeled as fuzzy intervals the optimality of solutions and the optimality of elements are characterized this characterization is performed in the setting of possibility theory a method of choosing a solution under uncertainty is also proposed

combinatorial optimization problems with uncertain costs and the owa criterion in this paper a class of combinatorial optimization problems with uncertain costs is discussed the uncertainty is modeled by specifying a discrete scenario set containing k distinct cost scenarios the ordered weighted averaging owa for short aggregation operator is applied to choose a solution some well known criteria used in decision making under uncertainty such as the maximum minimum average hurwicz and median are special cases of owa furthermore by using owa the traditional robust min max approach to combinatorial optimization problems with uncertain costs can be generalized the computational complexity and approximability of the problem of minimizing owa for the considered class of problems are investigated and some new positive and negative results in this area are provided these results remain valid for many basic problems such as network or resource allocation problems

continuum structural topology design with genetic algorithms the genetic algorithm ga an optimization technique based on the theory of natural selection is applied to structural topology design problems after reviewing the ga and previous research in structural topology optimization we describe a binary material void design representation that is encoded in ga chromosome data structures this representation is intended to approximate a material continuum as opposed to discrete truss structures four examples showing the broad utility of the approach and representation are then presented a fifth example suggests an alternate representation that allows continuously variable material density concluding discussion suggests recommended uses of the technique and describes ongoing and possible future work

ordered weighted average optimization in multiobjective spanning tree problem abstract multiobjective spanning tree problems are studied in this paper the ordered median objective function is used as an averaging operator to aggregate the vector of objective values of feasible solutions this leads to the ordered weighted average spanning tree problem a nonlinear combinatorial optimization problem different mixed integer linear programs are proposed based on the most relevant minimum cost spanning tree models in the literature these formulations are analyzed and several enhancements presented their empirical performance is tested over a set of randomly generated benchmark instances the results of the computational experiments show that the choice of an appropriate formulation allows to solve larger instances with more objectives than those previously solved in the literature

minimax regret spanning arborescences under uncertain costs abstract the paper considers a classical optimization problem on a network whose arc costs are partially known it is assumed that an interval estimate is given for each arc cost and no further information about the statistical distribution of the truth value of the arc cost is known in this context given a spanning arborescence in the network its cost can take on different values according to the choice of each individual arc cost that is according to the different cost scenarios we analyze the problem of finding which spanning arborescence better approaches the optimal one under each possible scenario the minimax regret criterion is proposed in order to obtain such a robust solution to the problem in the paper it is shown that a greedy type algorithm can compute an optimal solution of this problem on acyclic networks for general networks the problem becomes np hard in this case the special structure of the optimization problem allows us to design a bounding process for the optimum value that will result in a heuristic algorithm described at the end of the paper

minmax regret combinatorial optimization problems with ellipsoidal uncertainty sets we consider robust counterparts of uncertain combinatorial optimization problems where the difference to the best possible solution over all scenarios is to be minimized such minmax regret problems are typically harder to solve than their nominal non robust counterparts while current literature almost exclusively focuses on simple uncertainty sets that are either finite or hyperboxes we consider problems with more flexible and realistic ellipsoidal uncertainty sets we present complexity results for the unconstrained combinatorial optimization problem the shortest path problem and the minimum spanning tree problem to solve such problems two types of cuts are introduced and compared in a computational experiment

genetic search an approach to the nonconvex optimization problem principles of genetics and natural selection are adapted into a search procedure for function optimization such methods are based on a randomized selection from that restricted region of the design space that yields and improvement in the objective function an implementation of the approach to a class of problems in structural optimization with demonstrated nonconvexities or disjointness is discussed in the paper the principal drawback of the method is an increase in function evaluations necessary to locate an optimum possible strategies to overcome this limitation are presented

an enhanced genetic algorithm for structural topology optimization genetic algorithms gas have become a popular optimization tool for many areas of research and topology optimization an effective design tool for obtaining efficient and lighter structures in this paper a versatile robust and enhanced ga is proposed for structural topology optimization by using problem specific knowledge the original discrete black and white 0 1 problem is directly solved by using a bit array representation method to address the related pronounced connectivity issue effectively the four neighbourhood connectivity is used to suppress the occurrence of checkerboard patterns a simpler version of the perimeter control approach is developed to obtain a well posed problem and the total number of hinges of each individual is explicitly penalized to achieve a hinge free design to handle the problem of representation degeneracy effectively a recessive gene technique is applied to viable topologies while unusable topologies are penalized in a hierarchical manner an efficient fem based function evaluation method is developed to reduce the computational cost a dynamic penalty method is presented for the ga to convert the constrained optimization problem into an unconstrained problem without the possible degeneracy with all these enhancements and appropriate choice of the ga operators the present ga can achieve significant improvements in evolving into near optimum solutions and viable topologies with checkerboard free mesh independent and hinge free characteristics numerical results show that the present ga can be more efficient and robust than the conventional gas in solving the structural topology optimization problems of minimum compliance design minimum weight design and optimal compliant mechanisms design it is suggested that the present enhanced ga using problem specific knowledge can be a powerful global search tool for structural topology optimization copyright 2005 john wiley sons ltd

motion estimation via dynamic vision estimating the three dimensional motion of an object from a sequence of projections is of paramount importance in a variety of applications in control and robotics such as autonomous navigation manipulation servo tracking docking planning and surveillance although visual motion estimation is an old problem the first formulations date back to the beginning of the century only recently have tools from nonlinear systems estimation theory hinted at acceptable solutions in this paper the authors formulate the visual motion estimation problem in terms of identification of nonlinear implicit systems with parameters on a topological manifold and propose a dynamic solution either in the local coordinates or in the embedding space of the parameter manifold such a formulation has structural advantages over previous recursive schemes since the estimation of motion is decoupled from the estimation of the structure of the object being viewed and therefore it is possible to handle occlusions in a principled way

the computational complexity of the relative robust shortest path problem with interval data abstract the paper deals with the relative robust shortest path problem in a directed arc weighted graph where arc lengths are specified as intervals containing possible realizations of arc lengths the complexity status of this problem has been unknown in the literature we show that the problem is np hard

a probabilistic model for minmax regret in combinatorial optimization in this paper we propose a new probabilistic model for minimizing the anticipated regret in combinatorial optimization problems with distributional uncertainty in the objective coefficients the interval uncertainty representation of data is supplemented with information on the marginal distributions as a decision criterion we minimize the worst case conditional value at risk of regret the proposed model includes the interval data minmax regret model as a special case for the class of combinatorial optimization problems with a compact convex hull representation a polynomial sized mixed integer linear program is formulated when a the range and mean are known and b the range mean and mean absolute deviation are known and a mixed integer second order cone program is formulated when c the range mean and standard deviation are known for the subset selection problem of choosing k elements of maximum total weight out of a set of n elements the probabilistic regret model is shown to be solvable in polynomial time in the instances a and b above this extends the current known polynomial complexity result for minmax regret subset selection with range information only

combined gradient stochastic optimization with negative circular masks for large deformation topologies summary topology optimization of large deformation two dimensional continua is presented using a combined gradient stochastic search with negative circular masks the possibility of generating perfect black and white topologies is explored while attaining the efficiency of first order and second order search algorithms the design region is modeled with honeycomb tessellation to thwart the known connectivity singularities such as the checkerboards and point flexures mask shrinkage is incorporated for ease in density transition between gradient and stochastic steps notches at continuum boundaries are moderated through multiple use of a simple boundary smoothing method a neo hookean elasticity model is employed to simulate the material nonlinearities in large displacement continua with examples on stiff beams and large deformation compliant mechanisms it is illustrated that perfectly binary connected and smooth topologies can be obtained within a few hundred design evaluations copyright 2012 john wiley sons ltd

robust submodular observation selection in many applications one has to actively select among a set of expensive observations before making an informed decision for example in environmental monitoring we want to select locations to measure in order to most effectively predict spatial phenomena often we want to select observations which are robust against a number of possible objective functions examples include minimizing the maximum posterior variance in gaussian process regression robust experimental design and sensor placement for outbreak detection in this paper we present the submodular saturation algorithm a simple and efficient algorithm with strong theoretical approximation guarantees for cases where the possible objective functions exhibit submodularity an intuitive diminishing returns property moreover we prove that better approximation algorithms do not exist unless np complete problems admit efficient algorithms we show how our algorithm can be extended to handle complex cost functions incorporating non unit observation cost or communication and path costs we also show how the algorithm can be used to near optimally trade off expected case e g the mean square prediction error in gaussian process regression and worst case e g maximum predictive variance performance we show that many important machine learning problems fit our robust submodular observation selection formalism and provide extensive empirical evaluation on several real world problems for gaussian process regression our algorithm compares favorably with state of the art heuristics described in the geostatistics literature while being simpler faster and providing theoretical guarantees for robust experimental design our algorithm performs favorably compared to sdp based algorithms c 2008 andreas krause h brendan mcmahan carlos guestrin and anupam gupta krause mcmahan guestrin and gupta

deterministic risk control for cost effective network connections this paper considers the minimum connection problem in networks with uncertain data in such a network it is assumed that one can establish a link e by paying a cost c e in a given interval c e c e while taking a risk c e c e c e c e of link failure we develop polynomial time algorithms for minimum cost network connection with paths or spanning trees under risk sum constraints

optimization criteria and geometric algorithms for motion and structure estimation prevailing efforts to study the standard formulation of motion and structure recovery have recently been focused on issues of sensitivity and robustness of existing techniques while many cogent observations have been made and verified experimentally many statements do not hold in general settings and make a comparison of existing techniques difficult with an ultimate goal of clarifying these issues we study the main aspects of motion and structure recovery the choice of objective function optimization techniques and sensitivity and robustness issues in the presence of noise we clearly reveal the relationship among different objective functions such as normalized epipolar constraints reprojection error or triangulation all of which can be unified in a new optimal triangulation procedure regardless of various choices of the objective function the optimization problems all inherit the same unknown parameter space the so called essential manifold based on recent developments of optimization techniques on riemannian manifolds in particular on stiefel or grassmann manifolds we propose a riemannian newton algorithm to solve the motion and structure recovery problem making use of the natural differential geometric structure of the essential manifold we provide a clear account of sensitivity and robustness of the proposed linear and nonlinear optimization techniques and study the analytical and practical equivalence of different objective functions the geometric characterization of critical points and the simulation results clarify the difference between the effect of bas relief ambiguity rotation and translation confounding and other types of local minima this leads to consistent interpretations of simulation results over a large range of signal to noise ratio and variety of configurations

on accuracy robustness and tolerances in vector boolean optimization a boolean programming problem with a finite number of alternatives where initial coefficients costs of linear payoff functions are subject to perturbations is considered we define robust solution as a feasible solution which for a given set of realizations of uncertain parameters guarantees the minimum value of the worst case relative regret among all feasible solutions for the pareto optimality principle an appropriate definition of the worst case relative regret is specified it is shown that this definition is closely related to the concept of accuracy function being recently intensively studied in the literature we also present the concept of robustness tolerances of a single cost vector the tolerance is defined as the maximum level of perturbation of the cost vector which does not destroy the solution robustness we present formulae allowing the calculation of the robustness tolerance obtained for some initial costs the results are illustrated with several numerical examples

nonlinear mean shift over riemannian manifolds the original mean shift algorithm is widely applied for nonparametric clustering in vector spaces in this paper we generalize it to data points lying on riemannian manifolds this allows us to extend mean shift based clustering and filtering techniques to a large class of frequently occurring non vector spaces in vision we present an exact algorithm and prove its convergence properties as opposed to previous work which approximates the mean shift vector the computational details of our algorithm are presented for frequently occurring classes of manifolds such as matrix lie groups grassmann manifolds essential matrices and symmetric positive definite matrices applications of the mean shift over these manifolds are shown

restricted robust uniform matroid maximization under interval uncertainty for the problem of selecting p items with interval objective function coefficients so as to maximize total profit we introduce the r restricted robust deviation criterion and seek solutions that minimize the r restricted robust deviation this new criterion increases the modeling power of the robust deviation minmax regret criterion by reducing the level of conservatism of the robust solution it is shown that r restricted robust deviation solutions can be computed efficiently results of experiments and comparisons with absolute robustness robust deviation and restricted absolute robustness criteria are reported

some properties of the e matrix in two view motion estimation in the eight point linear algorithm for determining 3d motion structure from two perspective views using point correspondences the e matrix plays a central role the e matrix is defined as a skew symmetrical matrix containing the translation components postmultiplied by a rotation matrix the authors show that a necessary and sufficient condition for a 3 3 matrix to be so decomposable is that one of its singular values is zero and the other two are equal several other forms of this property are presented some applications are briefly described

a benders decomposition approach for the robust spanning tree problem with interval data abstract the robust spanning tree problem is a variation motivated by telecommunications applications of the classic minimum spanning tree problem in the robust spanning tree problem edge costs lie in an interval instead of having a fixed value interval numbers model uncertainty about the exact cost values a robust spanning tree is a spanning tree whose total cost minimizes the maximum deviation from the optimal spanning tree over all realizations of the edge costs this robustness concept is formalized in mathematical terms and is used to drive optimization this paper describes a new exact method based on benders decomposition for the robust spanning tree problem with interval data computational results highlight the efficiency of the new method which is shown to be very fast on all the benchmarks considered and in particular on those that were harder to solve for the methods previously known

essential matrix estimation using gauss newton iterations on a manifold a novel approach for essential matrix estimation is presented this being a key task in stereo vision processing we estimate the essential matrix from point correspondences between a stereo image pair assuming that the internal camera parameters are known the set of essential matrices forms a smooth manifold and a suitable cost function can be defined on this manifold such that its minimum is the desired essential matrix we seek a computationally efficient optimization scheme towards meeting the demands of on line processing of video images our work extends and improves the earlier research by ma et al who proposed an intrinsic riemannian newton method for essential matrix computations in contrast to ma et al we propose three gauss newton type algorithms that have improved convergence properties and reduced computational cost the first one is based on a novel intrinsic newton method using the normal riemannian metric on the manifold consisting of all essential matrices the other two methods are newton like methods that are more efficient from a numerical point of view local quadratic convergence of the algorithms is shown based on a careful analysis of the underlying geometry of the problem

optimality conditions of the hybrid cellular automata for structural optimization the hybrid cellular automaton method has been successfully applied to topology optimization using a uniform strain energy density distribution approach in this work a new set of design rules is derived from the first order optimality conditions of a multi objective problem in this new formulation the final topology is derived to minimize both mass and strain energy in the hybrid cellular automaton algorithm local design rules based on the cellular automaton paradigm are used to efficiently drive the design to optimality in addition to the control based techniques previously introduced a new ratio technique is derived in this investigation this work also compares the performance of the control strategies and the ratio technique

an improved algorithm for selecting p items with uncertain returns according to the minmax regret criterion we consider the problem of selecting a subset of p investments of maximum total return out of a set of n available investments with uncertain returns where uncertainty is represented by interval estimates for the returns and the minmax regret objective is used we develop an algorithm that solves this problem in o min p n p n time this improves the previously known complexity o min p n p 2n

using bounds on the data in linear programming the tolerance approach to sensitivity analysis in contrast to traditional sensitivity analysis in linear programming the tolerance approach considers simultaneous and independent variations in a number of parameters a primary focus of this approach is to determine a maximum tolerance percentage for selected right hand side terms in which the same basis is optimal as long as each term is accurate to within that percentage of its estimated value similarly the approach yields a maximum tolerance percentage for selected objective function coefficients this paper shows how the tolerance approach can exploit information on the range of possible values over which terms and coefficients can vary to yield larger maximum tolerance percentages

optimal design in elasticity and plasticity a direct weight minimization subject to compliance constraints or plastic yielding constraints leads to a non convex variational problem both the theoretical and the numerical analysis are unsatisfactory the minimum weight is not achieved by any design and the approximate designs oscillate as the element mesh is refined we look for equivalent relaxed problems with the same minima they come from allowing composite materials constructed in an optimal way from the original materials the constructions are different for elasticity and plasticity but surprisingly the final relaxed problems are in some cases the same

strong formulations of robust mixed 0 1 programming we introduce strong formulations for robust mixed 0 1 programming with uncertain objective coefficients we focus on a polytopic uncertainty set described by a budget constraint for allowed uncertainty in the objective coefficients we show that for a robust 0 1 problem there is an tight linear programming formulation with size polynomial in the size of an tight linear programming formulation for the nominal 0 1 problem we give extensions to robust mixed 0 1 programming and present computational experiments with the proposed formulations

camera displacement via constrained minimization of the algebraic error this paper proposes a new approach to estimate the camera displacement of stereo vision systems via minimization of the algebraic error over the essential matrices manifold the proposed approach is based on the use of homogeneous forms and linear matrix inequality lmi optimizations and has the advantages of not presenting local minima and not introducing approximations of nonlinear terms numerical investigations carried out with both synthetic and real data show that the proposed approach provides significantly better results than svd methods as well as minimizations of the algebraic error over the essential matrices manifold via both gradient descent and simplex search algorithms

semidefinite relaxations of robust binary least squares under ellipsoidal uncertainty sets the problem of finding the least squares solution s to a system of equations hs y is considered when s is a vector of binary variables and the coefficient matrix h is unknown but of bounded uncertainty similar to previous approaches to robust binary least squares we explore the potential of a min max design with the aim to provide solutions that are less sensitive to the uncertainty in h we concentrate on the important case of ellipsoidal uncertainty i e the matrix h is assumed to be a deterministic unknown quantity which lies in a given uncertainty ellipsoid the resulting problem is np hard yet amenable to convex approximation techniques starting from a convenient reformulation of the original problem we propose an approximation algorithm based on semidefinite relaxation that explicitly accounts for the ellipsoidal uncertainty in the coefficient matrix next we show that it is possible to construct a tighter relaxation by suitably changing the description of the feasible region of the problem and formulate an approximation algorithm that performs better in practice interestingly both relaxations are derived as lagrange bidual problems corresponding to the two equivalent problem reformulations the strength of the proposed tightened relaxation is demonstrated by pertinent simulations

camera pose estimation via projective newton optimization on the manifold determining the pose of a moving camera is an important task in computer vision in this paper we derive a projective newton algorithm on the manifold to refine the pose estimate of a camera the main idea is to benefit from the fact that the 3 d rigid motion is described by the special euclidean group which is a riemannian manifold the latter is equipped with a tangent space defined by the corresponding lie algebra this enables us to compute the optimization direction i e the gradient and the hessian at each iteration of the projective newton scheme on the tangent space of the manifold then the motion is updated by projecting back the variables on the manifold itself we also derive another version of the algorithm that employs homeomorphic parameterization to the special euclidean group we test the algorithm on several simulated and real image data sets compared with the standard newton minimization scheme we are now able to obtain the full numerical formula of the hessian with a 60 decrease in computational complexity compared with levenberg marquardt the results obtained are more accurate while having a rather similar complexity

the minmax regret permutation flow shop problem with two jobs abstract we consider the makespan minimizing permutation flow shop problem with 2 jobs and m machines and uncertain processing times of operations for each operation an uncertainty interval is given and it is assumed that the processing time of each operation can take on any value from the corresponding uncertainty interval regardless of the values taken by processing times of other operations for the problem of finding a minmax regret solution we present an o m algorithm

on a constant factor approximation for minmax regret problems using a symmetry point scenario abstract in order to find a robust solution under an unknown linear cost function it will be considered the minmax regret criterion it is assumed the vector of costs can take on values from a given uncertainty set the resulting optimization model has been extensively analyzed in the literature when the uncertain costs are modeled by closed intervals unfortunately except for rare applications this problem has np hard complexity which has led to the appearance of approximated methods seeking for good solutions in a short computational time the most recent approaches solve the deterministic version of the optimization problem under the midpoint scenario that realization in which each cost takes the midpoint of the closed interval proceeding in this way it can be obtained a 2 approximation for a wide class of minmax regret optimization models in this paper it is shown how this approach can be extended to a new class of minmax regret problems including more general uncertainty sets

explicit reformulations for robust optimization problems with general uncertainty sets we consider a rather general class of mathematical programming problems with data uncertainty where the uncertainty set is represented by a system of convex inequalities we prove that the robust counterparts of this class of problems can be reformulated equivalently as finite and explicit optimization problems moreover we develop simplified reformulations for problems with uncertainty sets defined by convex homogeneous functions our results provide a unified treatment of many situations that have been investigated in the literature and are applicable to a wider range of problems and more complicated uncertainty sets than those considered before the analysis in this paper makes it possible to use existing continuous optimization algorithms to solve more complicated robust optimization problems the analysis also shows how the structure of the resulting reformulation of the robust counterpart depends both on the structure of the original nominal optimization problem and on the structure of the uncertainty set

choosing robust solutions in discrete optimization problems with fuzzy costs in this paper a wide class of discrete optimization problems which can be formulated as a 0 1 linear programming problem is discussed it is assumed that the objective function costs are not precisely known this uncertainty is modeled by specifying a finite set of fuzzy scenarios under every fuzzy scenario the costs are given as fuzzy intervals possibility theory is then applied to chose a solution in such a problem and mixed integer linear programming models are proposed to compute this solution

structural design optimization subjected to uncertainty using fat bezier curve the success of genetic algorithms in structural design optimization depends largely on the geometric representation used in this work a geometric representation scheme using fat bezier curve is proposed and evaluated to be efficient and effective in producing good results via a structure design problem subjected to uncertainty this scheme facilitates the transmission of topological and shape characteristics across generations in the evolutionary process and amplifies the representation variability a hybrid genetic algorithm coupled with the scheme to tackle structure topology optimization under uncertainty is also presented the proposed hybrid algorithm integrates a simple local search strategy as the worst case scenario technique of anti optimization with a constrained multi objective genetic algorithm numerical results are also presented

on recoverable and two stage robust selection problems with budgeted uncertainty in this paper the problem of selecting p out of n available items is discussed such that their total cost is minimized we assume that costs are not known exactly but stem from a set of possible outcomes robust recoverable and two stage models of this selection problem are analyzed in the two stage problem up to p items is chosen in the first stage and the solution is completed once the scenario becomes revealed in the second stage in the recoverable problem a set of p items is selected in the first stage and can be modified by exchanging up to k items in the second stage after a scenario reveals we assume that uncertain costs are modeled through bounded uncertainty sets i e the interval uncertainty sets with an additional linear budget constraint in their discrete and continuous variants polynomial algorithms for recoverable and two stage selection problems with continuous bounded uncertainty and compact mixed integer formulations in the case of discrete bounded uncertainty are constructed

on the complexity of a class of combinatorial optimization problems with uncertainty we consider a robust minmax regret version of the problem of selecting p elements of minimum total weight out of a set of m elements with uncertainty in weights of the elements we present a polynomial algorithm with the order of complexity o min p m p 2m for the case where uncertainty is represented by means of interval estimates for the weights we show that the problem is np hard in the case of an arbitrary finite set of possible scenarios even if there are only two possible scenarios this is the first known example of a robust combinatorial optimization problem that is np hard in the case of scenario represented uncertainty but is polynomially solvable in the case of the interval representation of uncertainty

structural topology design optimization using genetic algorithms with a bit array representation abstract in this paper a bit array representation method for structural topology optimization using the genetic algorithm ga is implemented the importance of structural connectivity in a design is further emphasized by considering the total number of connected objects of each individual explicitly in an equality constraint function to evaluate the constrained objective function deb s constraint handling approach is further developed to ensure that feasible individuals are always better than infeasible ones in the population to improve the efficiency of the ga a violation penalty method is proposed to drive the ga search towards the topologies with higher structural performance less unusable material and fewer separate objects in the design domain an identical initialization method is also proposed to improve the ga performance in dealing with problems with long narrow design domains numerical results of structural topology optimization problems of minimum weight and minimum compliance designs show the success of this bit array representation method and suggest that the ga performance can be significantly improved by handling the design connectivity properly