Academic literature on the topic 'Lagrange optimisation'

The set-valued optimisation problem with constraints is considered in the sense of super efficiency in locally convex linear topological spaces. Under the assumption of nearly cone-subconvexlikeness, by applying the separation theorem for convex sets, Kuhn-Tucker and Lagrange necessary conditions for the set-valued optimisation problem to attain its super efficient solutions are obtained. Also, Kuhn-Tucker and Lagrange sufficient conditions are derived. Finally two kinds of unconstrained programs equivalent to set-valued optimisation problems are established.

Methodical aspects of the reliability-based structural optimisation using stochastic quasigradient methods are considered. For an example of the simply supported reinforced concrete beam, the employment of the Lagrange multiplier method that belongs to the class of stochastic quasigradient methods is demonstrated. The classical optimum design goal to minimise structural cost or weight under the constraint on the structural failure probability is taken for consideration. Optimisation problems solved with the Lagrangemultiplier method are formulated in form of general stochastic programming problem. The mathematical expectation of the concrete volume reduced with respect to the in-place cost of the beam materials is taken as the objective function. Constraint function is the limitation placed on the beam failure probability. The beam is considered as a series structural system. Values of the prescribed allowable failure probability belongs to the interval in which the estimation of the failure probabilities by the simple Monte-Carlomethod is possible with an acceptable confidence. The time-independent case as well as the time-dependent one is considered in the optimisation problems. The generalisation on the time-dependent case is undertaken through the introduction into the constraint function of the quasi-linear distribution law of the random variables. In the time-dependent case, the objective function is associated with beginning and the constraint function with end of the service period. An expression of the stochastic gradient based on the differentiation under the integral sign is used for calculations with the Lagrange multiplier method. The stochastic gradient used is computationally more effective in comparison with stochastic finite-difference formulae usual in stochastic quasigradient methods because it requires only one computation of the structure in search iteration of the optimisation process. Three rules based on statistical argumentation are used for the stopping of the seat according to the procedure of the Lagrange multiplier method. The optimising of the beam shows that the Lagrange multiplier method is applicable for the optimal design of structures in that cases when the structural reliability can be estimated by means of the simple Monte-Carlo method. Additional research is needed for integration in the Lagrange multiplier method of statistical simulation techniques for the estimation of small structural failure probabilities.

The subgradient, under strict efficiency, of a set-valued mapping is developed, and the existence of the subgradient is proved. Optimality conditions in terms of Lagrange multipliers for a strictly efficient point are established in the general case and in the case with ic-cone-convexlike data.

AbstractIn this paper, Antczak's η-approximation approach is used to prove the equivalence between optima of multiobjective programming problems and the η-saddle points of the associated η-approximated vector optimisation problems. We introduce an η-Lagrange function for a constructed η-approximated vector optimisation problem and present some modified η-saddle point results. Furthermore, we construct an η-approximated Mond-Weir dual problem associated with the original dual problem of the considered multiobjective programming problem. Using duality theorems between η-approximation vector optimisation problems and their duals (that is, an η-approximated dual problem), various duality theorems are established for the original multiobjective programming problem and its original Mond-Weir dual problem.

The majority of internet traffic is video content. This drives the demand for video compression in order to deliver high quality video at low target bitrates. This paper investigates the impact of adjusting the rate distortion equation on compression performance. An constant of proportionality, k, is used to modify the Lagrange multiplier used in H.265 (HEVC). Direct optimisation methods are deployed to maximise BD-Rate improvement for a particular clip. This leads to up to 21% BD-Rate improvement for an individual clip. Furthermore we use a more realistic corpus of material provided by YouTube. The results show that direct optimisation using BD-rate as the objective function can lead to further gains in bitrate savings that are not available with previous approaches.

AbstractThis paper addresses the motion planning problem of nonholonomic robotic systems. The system’s kinematics are described by a driftless control system with output. It is assumed that the control functions are represented in a parametric form, as truncated orthogonal series. A new motion planning algorithm is proposed based on the solution of a Lagrange-type optimisation problem stated in the linear approximation of the parametrised system. Performance of the algorithm is illustrated by numeric computations for a motion planning problem of the rolling ball.

Abstract A numerical procedure, which utilises polynomial power series expansions for the optimisation of multipanel wing structures in idealised critical flutter conditions, is introduced and developed. It arises from the Rayleigh-Ritz method and employes trial polynomial describing functions both for the flexural displacement and for the thickness variation over the multipanel surface. An idealised structural plate model, according to the Kirchhoff’s theory, together with a linearised supersonic aerodynamic approach, are supposed. The classical Euler-Lagrange optimality criterion, based on variational principles, has been utilised for the optimisation operations, where by imposing the stationary conditions of the Lagrangian functional expression, a nonlinear algebraic equations system is obtained, whose solution is found by an appropriate algorithm. By utilising an iterative process it is possible to reach the reference structure critical conditions, with an optimised thickness distribution throughout the multipanel surface. The final part of the work consists in searching the minimum weight of the multipanel planform wing structure with optimised thickness profile vs the flutter frequency, considered as a variable imput parameter, for fixed flutter speed and equal to the critical one of the reference uniform structure.

"This thesis investigates several non-linear analogues of Lagrange functions in the hope of answering the question 'Is it possible to generalise Lagrange functions such that they may be applied to a range of nonconvex objective problems?' The answer to this question is found to be yes for a particular class of optimization problems. Furthermore the thesis asserts that in derivative free optimization the general schema which is most theoretically and practically appealing involves the reformulation of both objective and constraint functions, whilst the least practically successful approach for everything but the most simple convex case is the augmented Lagrangian approach."
Doctor of Philosophy

"This thesis investigates several non-linear analogues of Lagrange functions in the hope of answering the question 'Is it possible to generalise Lagrange functions such that they may be applied to a range of nonconvex objective problems?' The answer to this question is found to be yes for a particular class of optimization problems. Furthermore the thesis asserts that in derivative free optimization the general schema which is most theoretically and practically appealing involves the reformulation of both objective and constraint functions, whilst the least practically successful approach for everything but the most simple convex case is the augmented Lagrangian approach."
Doctor of Philosophy

Depuis son introduction, la programmation mathématique à deux niveaux suscite un intérêt toujours croissant. En effet, vu ses applications dans une multitude de problèmes concrets (problèmes de gestion, planification économique, chimie, sciences environnementales,...), beaucoup de recherches ont été effectuées afin de contribuer à la résolution de cette classe de problèmes. Cette thèse est consacrée à l'étude de quelques classes de problèmes d'optimisation à deux niveaux, à savoir, les problèmes à deux niveaux forts, les problèmes à deux niveaux forts-faibles et les problèmes à deux niveaux semi-vectoriels. Le premier chapitre est consacré aux rappels de quelques définitions et résultats de topologie et d'analyse convexe que nous avons utilisé dans la suite. Dans le deuxième chapitre, nous avons rappelé quelques résultats théoriques et algorithmiques établis dans la littérature pour la résolution de quelques classes de problèmes d'optimisation à deux niveaux. Le troisième chapitre est consacré à l'étude d'un problème à deux niveaux fort-faible (SWBL). Vu la difficulté que présente cette classe de problèmes dans l'étude de l'existence de solutions, et afin de donner de nouvelles perspectives à leur résolution, nous avons procédé à une régularisation du problème. Sous des conditions suffisantes et via cette régularisation, nous avons montré que le problème (SWBL) admet au moins une solution. Dans le quatrième chapitre, nous avons donné une approche de dualité à un problème d'optimisation à deux niveaux fort (S). Cette approche est basée sur l'utilisation d'une régularisation et la dualité de Fenchel-Lagrange. En utilisant cette approche, nous avons donné des conditions nécessaires d'optimalité pour le problème (S). Enfin, des conditions suffisantes d'optimalité sont obtenues pour (S) sans utiliser l'approche. Une application concrète est donnée sur l'allocation de ressources. Dans le cinquième chapitre, nous avons étudié un problème à deux niveaux semi-vectoriel (SVBL). Pour ce problème, nous avons donné une approche de dualité en utilisant une régularisation, une scalarisation et la dualité de Fenchel-Lagrange. Puis, via cette approche et sous des hypothèses appropriées, nous avons donné des conditions nécessaires d'optimalité pour une classe de solutions du problème (SVBL). Finalement, des conditions suffisantes d'optimalité sont établies sont établies sans utiliser l'approche de dualité
Since its introduction, the class of tao-level programming problems has attracted increasing interest. Indeed, because of its applications in a multitude of concrete problems (management problems, economic planning, chemistry, environmental sciences,...), several researchers have been interested in the study of such class of problems. This thesis deals with the study of some classes of two-level optimization problems, namely, strong two-level problems, strong-weak two-level problems and semi-vectorial two-level problems. In the first chapter, we have recalled some definitions and results related to topology and convex analysis that we have used in our study. In the second chapter, we have discussed some theoretical and algorithmic results established in the literature for solving some classes of two-level optimization problems. The third chapter deals with strong-weak Stackelberg problems. As it is well-known, such a class of problems presents difficulties in its study concerning the existence of solutions. So that, for a strong-weak two-level optimization problem, we have first given a regularization. Then, via this regularization and under appropriate assumptions we have shown the existence of solutions to such a problem. This result generalizes the one given in the literature for weak Stackelberg problems. In the fourth chapter, we have given a duality approach for a strong two-level programming problem (S). The duality approach is based on the use of a regularization and the Fenchel-Lagrange duality. Then, via this approach, we have given necessary optimality conditions for (S). Finally, sufficient optimality conditions are given for the initial problem (S). An application to a two-level resource allocation problem is given. In the fifth chapter, we have considered a semivectorial two-level programming problem (SVBL) where the upper and lower levels are vectorial and scalar respectively. For such a problem, we have given a duality approach based on the use of a regularization, a scalarization and the Fenchel-Lagrange duality. Then, via this approach we have established necessary optimality conditions for (SVBL). Finally, we have given sufficient optimality conditions without using the duality approach

Transition to turbulence and flow control are studied by means of numerical simulations for different simple shear flows. Linear and non-linear optimisation methods using the Lagrange multiplier technique are employed. In the linear framework as objective function the standard disturbance kinetic energy is chosen and the constraints involve the linearised Navier–Stokes equations. We consider both the optimal initial condition leading to the largest disturbance energy growth at finite times and the optimal time-periodic forcing leading to the largest asymptotic response for the case of the flat plate boundary layer excluding the leading edge. The optimal disturbances for spanwise wavelengths of the order of the boundary layer thickness are streamwise vortices exploiting the lift-up mechanism to create streaks. For long spanwise wavelengths it is the Orr mechanism combined with the amplification of oblique wave packets that is responsible for the disturbance growth. Also linear optimal disturbances are computed around a leading edge and the effect of the geometry is considered. It is found that two-dimentional disturbances originating upstream, relative to the leading edge of the plate are inefficient at generating a viable disturbance, while three dimentional disturbances are more amplified. In the non-linear framework a new approach using ideas from non-equilibrium thermodynamics is developed. We determine the initial condition on the laminar/turbulent boundary closest to the laminar state. Starting from the general evolution criterion of non-equilibrium systems we propose a method to optimise the route to the statistically steady turbulent state, i.e. the state characterised by the largest entropy production. This is the first time information from the fully turbulent state is included in the optimisation procedure. The method is applied to plane Couette flow. We show that the optimal initial condition is localised in space for realistic flow domains, while the disturbance visits bent streaks before breakdown. Feedback control is applied to the bypass-transition scenario with high levels of free-stream turbulence. The flow is the flat-plate boundary layer. In this scenario low frequency perturbations enter the boundary layer and streamwise elongated disturbances emerge due to non-modal growth. The so-called streaky structures are growing in amplitude until they reach high enough energy levels and break down into turbulent spots via their secondary instability. When control is applied in the form of wall blowing and suction, the growth of the streaks is delayed, which implies a delay of the whole transition process. Additionally, a comparison with experimental work is performed demonstrating a remarkable agreement in the disturbance attenuation once the differences between the numerical and experimental setup are reduced. Open-loop control with wall travelling waves by means of blowing and suction is applied to a separating boundary layer. For downstream travelling waves we obtain a mitigation of the separation of the boundary layer while for upstream travelling waves a significant delay in the transition location accompanied by a modest reduction of the separated region.
QC 20110518

The Constrained Shortest Path Problem (CSPP) consists of finding the shortest path in a graph or network that satisfies one or more resource constraints. Without these constraints, the shortest path problem can be solved in polynomial time; with them, the CSPP is NP-hard and thus far no polynomial-time algorithms exist for solving it optimally. The problem arises in a number of practical situations. In the case of vehicle path planning, the vehicle may be an aircraft flying through a region with obstacles such as mountains or radar detectors, with an upper bound on the fuel consumption, the travel time or the risk of attack. The vehicle may be a submarine travelling through a region with sonar detectors, with a time or risk budget. These problems all involve a network which is a discrete model of the physical domain. Another example would be the routing of voice and data information in a communications network such as a mobile phone network, where the constraints may include maximum call delays or relay node capacities. This is a problem of current economic importance, and one for which time-sensitive solutions are not always available, especially if the networks are large. We consider the simplest form of the problem, large grid networks with a single side constraint, which have been studied in the literature. This thesis explores the application of Constraint Programming combined with Lagrange Relaxation to achieve optimal or near-optimal solutions of the CSPP. The following is a brief outline of the contribution of this thesis. Lagrange Relaxation may or may not achieve optimal or near-optimal results on its own. Often, large duality gaps are present. We make a simple modification to Dijkstra’s algorithm that does not involve any additional computational work in order to generate an estimate of path time at every node.We then use this information to constrain the network along a bisecting meridian. The combination of Lagrange Relaxation (LR) and a heuristic for filtering along the meridian provide an aggressive method for finding near-optimal solutions in a short time. Two network problems are studied in this work. The first is a Submarine Transit Path problem in which the transit field contains four sonar detectors at known locations, each with the same detection profile. The side constraint is the total transit time, with the submarine capable of 2 speeds. For the single-speed case, the initial LR duality gap may be as high as 30%. The first hybrid method uses a single centre meridian to constrain the network based on the unused time resource, and is able to produce solutions that are generally within 1% of optimal and always below 3%. Using the computation time for the initial Lagrange Relaxation as a baseline, the average computation time for the first hybrid method is about 30% to 50% higher, and the worst case CPU times are 2 to 4 times higher. The second problem is a random valued network from the literature. Edge costs, times, and lengths are uniform, randomly generated integers in a given range. Since the values given in the literature problems do not yield problems with a high duality gap, the values are varied and from a population of approximately 100,000 problems only the worst 200 from each set are chosen for study. These problems have an initial LR duality gap as high as 40%. A second hybrid method is developed, using values for the unused time resource and the lower bound values computed by Dijkstra’s algorithm as part of the LR method. The computed values are then used to position multiple constraining meridians in order to allow LR to find better solutions.This second hybrid method is able to produce solutions that are generally within 0.1% of optimal, with computation times that are on average 2 times the initial Lagrange Relaxation time, and in the worst case only about 5 times higher. The best method for solving the Constrained Shortest Path Problem reported in the literature thus far is the LRE-A method of Carlyle et al. (2007), which uses Lagrange Relaxation for preprocessing followed by a bounded search using aggregate constraints. We replace Lagrange Relaxation with the second hybrid method and show that optimal solutions are produced for both network problems with computation times that are between one and two orders of magnitude faster than LRE-A. In addition, these hybrid methods combined with the bounded search are up to 2 orders of magnitude faster than the commercial CPlex package using a straightforward MILP formulation of the problem. Finally, the second hybrid method is used as a preprocessing step on both network problems, prior to running CPlex. This preprocessing reduces the network size sufficiently to allow CPlex to solve all cases to optimality up to 3 orders of magnitude faster than without this preprocessing, and up to an order of magnitude faster than using Lagrange Relaxation for preprocessing. Chapter 1 provides a review of the thesis and some terminology used. Chapter 2 reviews previous approaches to the CSPP, in particular the two current best methods. Chapter 3 applies Lagrange Relaxation to the Submarine Transit Path problem with 2 speeds, to provide a baseline for comparison. The problem is reduced to a single speed, which demonstrates the large duality gap problem possible with Lagrange Relaxation, and the first hybrid method is introduced.Chapter 4 examines a grid network problem using randomly generated edge costs and weights, and introduces the second hybrid method. Chapter 5 then applies the second hybrid method to both network problems as a preprocessing step, using both CPlex and a bounded search method from the literature to solve to optimality. The conclusion of this thesis and directions for future work are discussed in Chapter 6.

The Constrained Shortest Path Problem (CSPP) consists of finding the shortest path in a graph or network that satisfies one or more resource constraints. Without these constraints, the shortest path problem can be solved in polynomial time; with them, the CSPP is NP-hard and thus far no polynomial-time algorithms exist for solving it optimally. The problem arises in a number of practical situations. In the case of vehicle path planning, the vehicle may be an aircraft flying through a region with obstacles such as mountains or radar detectors, with an upper bound on the fuel consumption, the travel time or the risk of attack. The vehicle may be a submarine travelling through a region with sonar detectors, with a time or risk budget. These problems all involve a network which is a discrete model of the physical domain. Another example would be the routing of voice and data information in a communications network such as a mobile phone network, where the constraints may include maximum call delays or relay node capacities. This is a problem of current economic importance, and one for which time-sensitive solutions are not always available, especially if the networks are large. We consider the simplest form of the problem, large grid networks with a single side constraint, which have been studied in the literature. This thesis explores the application of Constraint Programming combined with Lagrange Relaxation to achieve optimal or near-optimal solutions of the CSPP. The following is a brief outline of the contribution of this thesis. Lagrange Relaxation may or may not achieve optimal or near-optimal results on its own. Often, large duality gaps are present. We make a simple modification to Dijkstra’s algorithm that does not involve any additional computational work in order to generate an estimate of path time at every node.
We then use this information to constrain the network along a bisecting meridian. The combination of Lagrange Relaxation (LR) and a heuristic for filtering along the meridian provide an aggressive method for finding near-optimal solutions in a short time. Two network problems are studied in this work. The first is a Submarine Transit Path problem in which the transit field contains four sonar detectors at known locations, each with the same detection profile. The side constraint is the total transit time, with the submarine capable of 2 speeds. For the single-speed case, the initial LR duality gap may be as high as 30%. The first hybrid method uses a single centre meridian to constrain the network based on the unused time resource, and is able to produce solutions that are generally within 1% of optimal and always below 3%. Using the computation time for the initial Lagrange Relaxation as a baseline, the average computation time for the first hybrid method is about 30% to 50% higher, and the worst case CPU times are 2 to 4 times higher. The second problem is a random valued network from the literature. Edge costs, times, and lengths are uniform, randomly generated integers in a given range. Since the values given in the literature problems do not yield problems with a high duality gap, the values are varied and from a population of approximately 100,000 problems only the worst 200 from each set are chosen for study. These problems have an initial LR duality gap as high as 40%. A second hybrid method is developed, using values for the unused time resource and the lower bound values computed by Dijkstra’s algorithm as part of the LR method. The computed values are then used to position multiple constraining meridians in order to allow LR to find better solutions.
This second hybrid method is able to produce solutions that are generally within 0.1% of optimal, with computation times that are on average 2 times the initial Lagrange Relaxation time, and in the worst case only about 5 times higher. The best method for solving the Constrained Shortest Path Problem reported in the literature thus far is the LRE-A method of Carlyle et al. (2007), which uses Lagrange Relaxation for preprocessing followed by a bounded search using aggregate constraints. We replace Lagrange Relaxation with the second hybrid method and show that optimal solutions are produced for both network problems with computation times that are between one and two orders of magnitude faster than LRE-A. In addition, these hybrid methods combined with the bounded search are up to 2 orders of magnitude faster than the commercial CPlex package using a straightforward MILP formulation of the problem. Finally, the second hybrid method is used as a preprocessing step on both network problems, prior to running CPlex. This preprocessing reduces the network size sufficiently to allow CPlex to solve all cases to optimality up to 3 orders of magnitude faster than without this preprocessing, and up to an order of magnitude faster than using Lagrange Relaxation for preprocessing. Chapter 1 provides a review of the thesis and some terminology used. Chapter 2 reviews previous approaches to the CSPP, in particular the two current best methods. Chapter 3 applies Lagrange Relaxation to the Submarine Transit Path problem with 2 speeds, to provide a baseline for comparison. The problem is reduced to a single speed, which demonstrates the large duality gap problem possible with Lagrange Relaxation, and the first hybrid method is introduced.
Chapter 4 examines a grid network problem using randomly generated edge costs and weights, and introduces the second hybrid method. Chapter 5 then applies the second hybrid method to both network problems as a preprocessing step, using both CPlex and a bounded search method from the literature to solve to optimality. The conclusion of this thesis and directions for future work are discussed in Chapter 6.

La Séparation de Sources consiste à retrouver un jeu de M signaux indépendants appelés sources à partir de l'observation de N de leurs mélanges. Il existe de nombreuses méthodes de séparation qui sont pour la plupart basées sur les statistiques d'ordre supérieures. Ces méthodes permettent d'exploiter l'hypothèse d'indépendance des sources. Dans ce cadre nous avons considéré le cas de la séparation de sources basée sur l'optimisation de fonctions de contraste dans un cas de mélange linéaire purement spatial. Nous proposons d'abord deux nouvelles familles de contrastes regroupant comme cas particulier des contrastes existants. Puis nous déterminons la solution optimale dans le cas deux sources pour ces deux familles de contrastes. Cela nous permet finalement de proposer deux algorithmes, qui constituent alors deux généralisations d'algorithmes classiques. Ensuite, nous étudions l'optimisation d'un contraste sous contrainte, afin de proposer des algo¬rithmes ne nécessitant pas. Comme dans le cas précédent de blanchiment préalable des données. Deux familles de méthodes directes d'optimisation sous contrainte sont considérées : les méthodes duales et les méthodes directes. Cela nous permet en outre de développer des algorithmes utilisant les notions de Lagrangien, de pénalisation et de projection sur la contrainte. Des simulations informatiques illustrent le comportement des algorithmes proposés
Blind Source Separation aim to recover a set of M independent signals called sources from the observation of N mixtures. Several Source Separation methods exist, most of them are based on Higher Order Statistics. Those methods exploit the source independence hypothesis. Among them we consider the case of the source separation based on contrast function optimization in a spatial linear mixture case. We first propose two contrast families including as a particular case some existing contrasts. Then we determine the optimal solution in a two sources case for this couple of contrast families, thus proposing two algorithms which constitute two classic algorithm generalizations. Then, we study constrained contrast optimization in order to propose algorithms which don't need, as before, data pre-whitening. Two direct constrained optimization method families are considered : the dual methods and the direct methods. Thus we can develop algorithms using Lagrangian concept, penalization concept and a concept of projecting onto the constraint. Com¬puter simulations illustrate the behaviour of the algorithms

L'étude porte sur l'optimisation sous contraintes de formes en aérodynamique, avec application à la conception de lignes de nacelles de moteurs civils. Apres un rappel du contexte de l'étude et de sa place au sein des techniques de dessin automatique, on présente le formalisme B-spline retenu pour la paramétrisation de la forme. L'algorithme d'optimisation récursif quadratique est ensuite détaillé. L'étude d'un problème monodimensionnel simple permet de valider la technique de calcul du gradient par la résolution d'une équation adjointe, ainsi que ses variantes. Cette technique est ensuite étendue à l'analyse de sensibilité pour les équations d'Euler bidimensionnelles ou axisymètriques. Enfin, l'application à l'optimisation des nacelles civiles est présentée, au travers de plusieurs formulations possibles du problème, avec la prise en compte de contraintes tant géométriques qu'aérodynamiques

This chapter builds on the previous two chapters to tackle constrained systems, using Lagrangian mechanics and constrained variations. The first section deals with holonomic constraint equations using Lagrange multipliers; these can be used to reduce the number of coordinates until a linearly independent minimal set is obtained that describes a constraint surface within configuration space, so that Lagrange equations can be set up and solved. Motion is understood to be confined to a constraint submanifold. The variational formulation of non-holonomic constraints is then discussed to derive the vakonomic formulation. These erroneous equations are then compared to the central Lagrange equation, and the precise nature of the variations used in each formulation is investigated. The vakonomic equations are then presented in their Suslov form (Suslov–vakonomic form) in an attempt to reconcile the two approaches. In addition, the structure of biological membranes is framed as a constrained optimisation problem.

This chapter looks at the calculus of a function of two or more variables, which is the subject of partial differentiation. The partial derivative of a function is the rate of change of the function with respect to the distance in the direction of a particular coordinate axis and is symbolised with the sign ∂. The chapter spends time on the implicit function theorem, since it is relied upon heavily elsewhere in the text. Lagrange multipliers are used to solve constrained optimisation problems. Topics include critical points, the product rule, the chain rule, directional derivatives, hypersurfaces and Taylor’s theorem. In addition, the chapter discusses Jacobian matrices, the inverse function theorem, gradients, level sets and Hessian matrices.

This paper presents results from recent numerical experiments supported by theoretical arguments which indicate where are the limits of current optimization methods when applied to problems involving a large number of design variables as well as a large number of constraints, many of them being active. This is typical of optimal sizing problems with local stress constraints especially when composite materials are employed. It is shown that in both primal and dual methods the CPU time spent in the optimizer is related to the numerical effort needed to invert a symmetric positive definite matrix of size jact, jact being the effective number of active constraints, i.e. constraints associated with positive Lagrange multipliers. This CPU time varies with jact3. When the number m of constraints increases, jact has a tendency to grow, but there is a limit. Indeed another well known theoretical property is that the number of active constraints jact should not exceed the number of free primal variables iact, i.e. the number of variables that do not reach a lower or upper bound. This number iact is itself of course smaller than the real number of design variables n. This leads to the conclusion that for problems with many active constraints the CPU time could grow as fast as n3. With respect to m the increase in CPU time remains approximately linear. Some practical applications to real life industrial problems will be briefly shown: design optimisation of an aircraft composite wing with local buckling constraints and topology optimization of an engine pylon.