structural topology optimization with eigenvalues the paper considers different problem formulations of topology optimization of discrete or discretized structures with eigenvalues as constraints or as objective functions we study multiple load case formulations of minimum volume or weight minimum compliance problems and the problem of maximizing the minimal eigenvalue of the structure including the effect of nonstructural mass the paper discusses interrelations of the problems and in particular shows how solutions of one problem can be derived from solutions of the others moreover we present equivalent reformulations as semidefinite programming problems with the property that for the minimum volume and minimum compliance problem each local optimizer of these problems is also a global one this allows for the calculation of guaranteed global optimizers of the original problems through the use of modern solution techniques of semidefinite programming for the problem of maximization of the minimum eigenvalue we show how to verify the global optimality and present an algorithm for finding a tight approximation of a globally optimal solution numerical examples are provided for truss structures both academic and larger size examples illustrate the theoretical results achieved and demonstrate the practical use of this approach we conclude with an extension on multiple nonstructural mass conditions

nonlinear semidefinite programming sensitivity convergence and an application in passive reduced order modeling we consider the solution of nonlinear programs with nonlinear semidefiniteness constraints the need for an efficient exploitation of the cone of positive semidefinite matrices makes the solution of such nonlinear semidefinite programs more complicated than the solution of standard nonlinear programs this paper studies a sequential semidefinite programming ssp method which is a generalization of the well known sequential quadratic programming method for standard nonlinear programs we present a sensitivity result for nonlinear semidefinite programs and then based on this result we give a self contained proof of local quadratic convergence of the ssp method we also describe a class of nonlinear semidefinite programs that arise in passive reduced order modeling and we report results of some numerical experiments with the ssp method applied to problems in that class

on the r superlinear convergence of the kkt residuals generated by the augmented lagrangian method for convex composite conic programming due to the possible lack of primal dual type error bounds it was not clear whether the karush kuhn tucker kkt residuals of the sequence generated by the augmented lagrangian method alm for solving convex composite conic programming cccp problems converge superlinearly in this paper we resolve this issue by establishing the r superlinear convergence of the kkt residuals generated by the alm under only a mild quadratic growth condition on the dual of cccp with easy to implement stopping criteria for the augmented lagrangian subproblems this discovery may help to explain the good numerical performance of several recently developed semismooth newton cg based alm solvers for linear and convex quadratic semidefinite programming

robust solutions of linear programming problems contaminated with uncertain data optimal solutions of linear programming problems may become severely infeasible if the nominal data is slightly perturbed we demonstrate this phenomenon by studying 90 lps from the well known netlib collection we then apply the robust optimization methodology ben tal and nemirovski 1 3 el ghaoui et al 5 6 to produce robust solutions of the above lps which are in a sense immuned against uncertainty surprisingly for the netlib problems these robust solutions nearly lose nothing in optimality

loss of superlinear convergence for an sqp type method with conic constraints in this short note we consider a sequential quadratic programming sqp type method with conic subproblems and compare this method with a standard sqp method in which the conic constraint is linearized at each step for both approaches we restrict our attention to convex subproblems since these are easy to solve and guarantee a certain global descent property using the example of a simple nonlinear program nlp and its conic reformulation we show that the sqp method with conic subproblems displays a slower rate of convergence than standard sqp methods we then explain why an sqp subproblem that is based on a better approximation of the feasible set of the nlp results in a much slower algorithm

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

optimality conditions for nonconvex semidefinite programming this paper concerns nonlinear semidefinite programming problems for which no convexity assumptions can be made we derive first and second order optimality conditions analogous to those for nonlinear programming using techniques similar to those used in nonlinear programming we extend existing theory to cover situations where the constraint matrix is structurally sparse the discussion covers the case when strict complementarity does not hold the regularity conditions used are consistent with those of nonlinear programming in the sense that the conventional optimality conditions for nonlinear programming are obtained when the constraint matrix is diagonal

the scenario approach to robust control design this paper proposes a new probabilistic solution framework for robust control analysis and synthesis problems that can be expressed in the form of minimization of a linear objective subject to convex constraints parameterized by uncertainty terms this includes the wide class of np hard control problems representable by means of parameter dependent linear matrix inequalities lmis it is shown in this paper that by appropriate sampling of the constraints one obtains a standard convex optimization problem the scenario problem whose solution is approximately feasible for the original usually infinite set of constraints i e the measure of the set of original constraints that are violated by the scenario solution rapidly decreases to zero as the number of samples is increased we provide an explicit and efficient bound on the number of samples required to attain a priori specified levels of probabilistic guarantee of robustness a rich family of control problems which are in general hard to solve in a deterministically robust sense is therefore amenable to polynomial time solution if robustness is intended in the proposed risk adjusted sense

robust model predictive control via scenario optimization this paper discusses a novel probabilistic approach for the design of robust model predictive control mpc laws for discrete time linear systems affected by parametric uncertainty and additive disturbances the proposed technique is based on the iterated solution at each step of a finite horizon optimal control problem fhocp that takes into account a suitable number of randomly extracted scenarios of uncertainty and disturbances followed by a specific command selection rule implemented in a receding horizon fashion the scenario fhocp is always convex also when the uncertain parameters and disturbance belong to nonconvex sets and irrespective of how the model uncertainty influences the system s matrices moreover the computational complexity of the proposed approach does not depend on the uncertainty disturbance dimensions and scales quadratically with the control horizon the main result in this work is related to the analysis of the closed loop system under receding horizon implementation of the scenario fhocp and essentially states that the devised control law guarantees constraint satisfaction at each step with some a priori assigned probability p while the system s state reaches the target set either asymptotically or in finite time with probability at least p the proposed method may be a valid alternative when other existing techniques either deterministic or stochastic are not directly usable due to excessive conservatism or to numerical intractability caused by lack of convexity of the robust or chance constrained optimization problem

first and second order analysis of nonlinear semidefinite programs in this paper we study nonlinear semidefinite programming problems convexity duality and first order optimality conditions for such problems are presented a second order analysis is also given second order necessary and sufficient optimality conditions are derived finally sensitivity analysis of such programs is discussed

sufficient optimality conditions hold for almost all nonlinear semidefinite programs we derive a new genericity result for nonlinear semidefinite programming nlsdp namely almost all linear perturbations of a given nlsdp are shown to be nondegenerate here nondegeneracy for nlsdp refers to the transversality constraint qualification strict complementarity and second order sufficient condition due to the presence of the second order sufficient condition our result is a nontrivial extension of the corresponding results for linear semidefinite programs sdp from alizadeh et al math program 77 2 ser b 111 128 1997 the proof of the genericity result makes use of forsgren s derivation of optimality conditions for nlsdp in forsgren math program 88 1 ser a 105 128 2000 due to the latter approach the positive semidefiniteness of a symmetric matrix g x depending continuously on x is locally equivalent to the fact that a certain schur complement s x of g x is positive semidefinite this yields a reduced nlsdp by considering the new semidefinite constraint s x succeq 0 s x 0 instead of g x succeq 0 g x 0 while deriving optimality conditions for the reduced nlsdp the well known and often mentioned h term in the second order sufficient condition vanishes this allows us to access the proof of the genericity result for nlsdp

robust truss topology design via semidefinite programming we present and motivate a new model of the truss topology design problem where the rigidity of the resulting truss with respect both to given loading scenarios and small occasional loads is optimized it is shown that the resulting optimization problem is a semidefinite program we derive and analyze several equivalent reformulations of the problem and present illustrative numerical examples

complementarity and nondegeneracy in semidefinite programming primal and dual nondegeneracy conditions are defined for semidefinite programming given the existence of primal and dual solutions it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution the converses hold if strict complementarity is assumed primal and dual nondegeneracy assumptions do not imply strict complementarity as they do in lp the primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutions x and z this is in contrast with lp where nondegeneracy assumptions exactly determine the number of variables which are zero it is shown that primal and dual nondegeneracy and strict complementarity all hold generically numerical experiments suggest probability distributions for the ranks of x and z which are consistent with the nondegeneracy conditions

bilevel optimization on the structure of the feasible set we consider bilevel optimization from the optimistic point of view let the pair x y denote the variables the main difficulty in studying such problems lies in the fact that the lower level contains a global constraint in fact a point x y is feasible if y solves a parametric optimization problem l x in this paper we restrict ourselves to the special case that the variable x is one dimensional we describe the generic structure of the feasible set m moreover we discuss local reductions of the bilevel problem as well as corresponding optimality criteria finally we point out typical problems that appear when trying to extend the ideas to higher dimensional x dimensions this will clarify the high intrinsic complexity of the general generic structure of the feasible set m and corresponding optimality conditions for the bilevel problem u

on constraint sampling in the linear programming approach to approximate dynamic programming in the linear programming approach to approximate dynamic programming one tries to solve a certain linear program the alp that has a relatively small numberk of variables but an intractable numberm of constraints in this paper we study a scheme that samples and imposes a subset ofm

robust constrained model predictive control using linear matrix inequalities abstract the primary disadvantage of current design techniques for model predictive control mpc is their inability to deal explicitly with plant model uncertainty in this paper we present a new approach for robust mpc synthesis that allows explicit incorporation of the description of plant uncertainty in the problem formulation the uncertainty is expressed in both the time and frequency domains the goal is to design at each time step a state feedback control law that minimizes a worst case infinite horizon objective function subject to constraints on the control input and plant output using standard techniques the problem of minimizing an upper bound on the worst case objective function subject to input and output constraints is reduced to a convex optimization involving linear matrix inequalities lmis it is shown that the feasible receding horizon state feedback control design robustly stabilizes the set of uncertain plants several extensions such as application to systems with time delays problems involving constant set point tracking trajectory tracking and disturbance rejection which follow naturally from our formulation are discussed the controller design is illustrated with two examples

the exact feasibility of randomized solutions of uncertain convex programs many optimization problems are naturally delivered in an uncertain framework and one would like to exercise prudence against the uncertainty elements present in the problem in previous contributions it has been shown that solutions to uncertain convex programs that bear a high probability to satisfy uncertain constraints can be obtained at low computational cost through constraint randomization in this paper we establish new feasibility results for randomized algorithms specifically the exact feasibility for the class of the so called fully supported problems is obtained it turns out that all fully supported problems share the same feasibility properties revealing a deep kinship among problems of this class it is further proven that the feasibility of the randomized solutions for all other convex programs can be bounded based on the feasibility for the prototype class of fully supported problems the feasibility result of this paper outperforms previous bounds and is not improvable because it is exact for fully supported problems

a sequential convex semidefinite programming algorithm with an application to multiple load free material optimization a new method for the efficient solution of a class of convex semidefinite programming sdp problems is introduced the method extends the sequential convex programming scp concept to optimization problems with matrix variables the basic idea of the new method is to approximate the original optimization problem by a sequence of subproblems in which nonlinear functions defined in matrix variables are approximated by block separable convex functions the subproblems are semidefinite programs with a favorable structure which can be efficiently solved by existing sdp software the new method is shown to be globally convergent the article is concluded by a series of numerical experiments with free material optimization problems demonstrating the effectiveness of the generalized scp approach

probabilistically constrained linear programs and risk adjusted controller design the focal point of this paper is the probabilistically constrained linear program pclp and how it can be applied to control system design under risk constraints the pclp is the counterpart of the classical linear program where it is assumed that there is random uncertainty in the constraints and therefore the deterministic constraints are replaced by probabilistic ones it is shown that for a wide class of probability density functions called log concave symmetric densities the pclp is a convex program an equivalent formulation of the pclp is also presented which provides insight into numerical implementation this concept is applied to control system design it is shown how the results in this paper can be applied to the design of controllers for discrete time systems to obtain a closed loop system with a well defined risk of violating the so called property of superstability furthermore we address the problem of risk adjusted pole placement

sequential convex approximations to joint chance constrained programs a monte carlo approach when there is parameter uncertainty in the constraints of a convex optimization problem it is natural to formulate the problem as a joint chance constrained program jccp which requires that all constraints be satisfied simultaneously with a given large probability in this paper we propose to solve the jccp by a sequence of convex approximations we show that the solutions of the sequence of approximations converge to a karush kuhn tucker kkt point of the jccp under a certain asymptotic regime furthermore we propose to use a gradient based monte carlo method to solve the sequence of convex approximations

free material optimization with fundamental eigenfrequency constraints the goal of this paper is to formulate and solve free material optimization problems with constraints on the smallest eigenfrequency of the optimal structure a natural formulation of this problem as a linear semidefinite program turns out to be numerically intractable as an alternative we propose a new approach which is based on a nonlinear semidefinite low rank approximation of the semidefinite dual we introduce an algorithm based on this approach and analyze its convergence properties the article is concluded by numerical experiments proving the effectiveness of the new approach

exploiting special structure in semidefinite programming a survey of theory and applications abstract semidefinite programming sdp may be seen as a generalization of linear programming lp in particular one may extend interior point algorithms for lp to sdp but it has proven much more difficult to exploit structure in the sdp data during computation we survey three types of special structures in sdp data 1 a common chordal sparsity pattern of all the data matrices this structure arises in applications in graph theory and may also be used to deal with more general sparsity patterns in a heuristic way 2 low rank of all the data matrices this structure is common in sdp relaxations of combinatorial optimization problems and sdp approximations of polynomial optimization problems 3 the situation where the data matrices are invariant under the action of a permutation group or more generally where the data matrices belong to a low dimensional matrix algebra such problems arise in truss topology optimization particle physics coding theory computational geometry and graph theory we will give an overview of existing techniques to exploit these structures in the data most of the paper will be devoted to the third situation since it has received the least attention in the literature so far

convex approximations of chance constrained programs we consider a chance constrained problem where one seeks to minimize a convex objective over solutions satisfying with a given close to one probability a system of randomly perturbed convex constraints this problem may happen to be computationally intractable our goal is to build its computationally tractable approximation i e an efficiently solvable deterministic optimization program with the feasible set contained in the chance constrained problem we construct a general class of such convex conservative approximations of the corresponding chance constrained problem moreover under the assumptions that the constraints are affine in the perturbations and the entries in the perturbation vector are independent of each other random variables we build a large deviation type approximation referred to as bernstein approximation of the chance constrained problem this approximation is convex and efficiently solvable we propose a simulation based scheme for bounding the optimal value in the chance constrained problem and report numerical experiments aimed at comparing the bernstein and well known scenario approximation approaches finally we extend our construction to the case of ambiguous chance constrained problems where the random perturbations are independent with the collection of distributions known to belong to a given convex compact set rather than to be known exactly while the chance constraint should be satisfied for every distribution given by this set

generic optimality conditions for semialgebraic convex programs we consider linear optimization over a nonempty convex semialgebraic feasible region f semidefinite programming is an example if f is compact then for almost every linear objective there is a unique optimal solution lying on a unique active manifold around which f is partly smooth and the second order sufficient conditions hold perturbing the objective results in smooth variation of the optimal solution the active manifold consists locally of these perturbed optimal solutions it is independent of the representation of f and is eventually identified by a variety of iterative algorithms such as proximal and projected gradient schemes these results extend to unbounded sets f

robust model predictive control using tubes a form of feedback model predictive control mpc that overcomes disadvantages of conventional mpc but which has manageable computational complexity is presented the optimal control problem solved on line yields a tube and an associated piecewise affine control law that maintains the controlled trajectories in the tube despite uncertainty computational complexity is linear rather than exponential in horizon length asymptotic stability of the controlled system is established

probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra we consider stochastic programming problems with probabilistic constraints involving random variables with discrete distributions they can be reformulated as large scale mixed integer programming problems with knapsack constraints using specific properties of stochastic programming problems and bounds on the probability of the union of events we develop new valid inequalities for these mixed integer programming problems we also develop methods for lifting these inequalities these procedures are used in a general iterative algorithm for solving probabilistically constrained problems the results are illustrated with a numerical example

stochastic control with uncertain parameters via chance constrained control chance constrained methods handle problem uncertainty by formulating the constraints probabilistically and only guaranteeing their satisfaction up to a given violation level this paper extends existing chance constrained methods to handle both uncertainty in the system state as well as in the constraint parameters due to the imperfect knowledge of the system state caused by motion sensor and environment uncertainty the system constraints cannot be guaranteed to be satisfied and consequently must be considered probabilistically prior work has primarily only dealt with uncertainty in the system state or constraint parameters but not both simultaneously however in many motivating applications both frequently occur to handle this case previous approaches either approximate the distribution leading to a nonconvex optimization program or use sampling alone to represent the uncertainty which requires a large number of samples to accurately represent the distribution to address these limitations a novel hybrid method is proposed that uses both analytical functions and sampling to represent the uncertainty it is shown that under certain conditions the resulting optimization program using this hybrid representation is convex to check the convexity an efficient a priori sufficient condition is developed furthermore by using this hybrid representation this method drastically reduces the computational complexity over previous methods which is demonstrated through two examples consequently this method has the ability to enable real time stochastic control for the motivating applications

parameter identification and synchronization for uncertain network group with different structures abstract we design a novel synchronization technique to research the synchronization of network group constituted of uncertain networks with different structures based on lyapunov theorem the selection principles of the control inputs and the parameter identification laws of the networks are determined and synchronization conditions of the network group are obtained some numerical simulations are provided to verify the correctness and effectiveness of the synchronization technique we find that the network number the number of network nodes and network connections indeed will not affect the stability of synchronization of network group

multidisciplinary free material optimization we present a mathematical framework for the so called multidisciplinary free material optimization mdfmo problems a branch of structural optimization in which the full material tensor is considered as a design variable we extend the original problem statement by a class of generic constraints depending either on the design or on the state variables among the examples are local stress or displacement constraints we show the existence of optimal solutions for this generalized free material optimization fmo problem and discuss convergent approximation schemes based on the finite element method

solving bilevel programs with the kkt approach bilevel programs bl form a special class of optimization problems they appear in many models in economics game theory and mathematical physics bl programs show a more complicated structure than standard finite problems we study the so called kkt approach for solving bilevel problems where the lower level minimality condition is replaced by the kkt or the fj condition this leads to a special structured mathematical program with complementarity constraints we analyze the kkt approach from a generic viewpoint and reveal the advantages and possible drawbacks of this approach for solving bl problems numerically