primal dual interior point methods for semidefinite programming convergence rates stability and numerical results primal dual interior point path following methods for semidefinite programming are considered several variants are discussed based on newton s method applied to three equations primal feasibility dual feasibility and some form of centering condition the focus is on three such algorithms called the xz xz zx and q methods for the xz zx and q algorithms the newton system is well defined and its jacobian is nonsingular at the solution under nondegeneracy assumptions the associated schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption practical aspects are discussed including mehrotra predictor corrector variants and issues of numerical stability compared to the other methods considered the xz zx method is more robust with respect to its ability to step close to the boundary converges more rapidly and achieves higher accuracy

a combined phase i phase ii scaled potential algorithm for linear programming we develop an extension of the affinely scaled potential reduction algorithm which simultaneously obtains feasibility and optimality in a standard form linear program without the addition of any m terms the method and its lower bounding procedure are particularly simple compared with previous interior algorithms not requiring feasibility

polynomial convergence of primal dual algorithms for semidefinite programming based on the monteiro and zhang family of directions this paper establishes the polynomial convergence of the class of primal dual feasible interior point algorithms for semidefinite programming sdp based on the monteiro and zhang family of search directions in contrast to monteiro and zhang s work math programming 81 1998 pp 281 299 here no condition is imposed on the scaling matrix that determines the search direction we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima mizuno and yoshise math programming 44 1989 pp 1 26 and monteiro and adler math programming 44 1989 pp 27 41 and pp 43 66 and the predictor corrector algorithm of mizuno todd and ye math oper res 18 1993 pp 945 981 carry over into the context of sdp since the monteiro and zhang family of directions includes the alizadeh haeberly and overton direction we establish for the first time the polynomial convergence of algorithms based on this search direction

affine parameter dependent lyapunov functions and real parametric uncertainty this paper presents new tests to analyze the robust stability and or performance of linear systems with uncertain real parameters these tests are extensions of the notions of quadratic stability and performance where the fixed quadratic lyapunov function is replaced by a lyapunov function with affine dependence on the uncertain parameters admittedly with some conservatism the construction of such parameter dependent lyapunov functions can be reduced to a linear matrix inequality lmi problem and hence is numerically tractable these lmi based tests are applicable to constant or time varying uncertain parameters and are less conservative than quadratic stability in the case of slow parametric variations they also avoid the frequency sweep needed in real spl mu analysis and numerical experiments indicate that they often compare favorably with spl mu analysis for time invariant parameter uncertainty

the projective method for solving linear matrix inequalities numerous problems in control and systems theory can be formulated in terms of linear matrix inequalities lmi since solving an lmi amounts to a convex optimization problem such formulations are known to be numerically tractable however the interest in lmi based design techniques has really surged with the introduction of efficient interior point methods for solving lmis with a polynomial time complexity this paper describes one particular method called the projective method simple geometrical arguments are used to clarify the strategy and convergence mechanism of the projective algorithm a complexity analysis is provided and applications to two generic lmi problems feasibility and linear objective minimization are discussed

superlinear convergence of a symmetric primal dual path following algorithm for semidefinite programming this paper establishes the superlinear convergence of a symmetric primal dual path following algorithm for semidefinite programming sdp under the assumptions that the semidefinite program has a strictly complementary primal dual optimal solution and that the size of the central path neighborhood tends to zero the interior point algorithm considered here closely resembles the mizuno todd ye predictor corrector method for linear programming which is known to be quadratically convergent it is shown that when the iterates are well centered the duality gap is reduced superlinearly after each predictor step indeed if each predictor step is succeeded by r consecutive corrector steps then the predictor reduces the duality gap superlinearly with order 2 1 2 r the proof relies on a careful analysis of the central path for sdp it is shown that under the strict complementarity assumption the primal dual central path converges to the analytic center of the primal dual optimal solution set and the distance from any point on the central path to this analytic center is bounded by the duality gap

a potential reduction newton method for constrained equations extending our previous work t wang r d c monteiro and j s pang math programming 74 1996 pp 159 195 this paper presents a general potential reduction newton method for solving a constrained system of nonlinear equations a major convergence result for the method is established specializations of the method to a convex semidefinite program and a monotone complementarity problem in symmetric matrices are discussed strengthened convergence results are established in the context of these specializations

spatial boundary problem with the dirichlet neumann condition for a singular elliptic equation abstract in the present work we find an explicit solution of a boundary problem with the dirichlet neumann condition for elliptic equation with singular coefficients in a quarter of ball for this aim the method of green s function has been used since found green s function contains a hypergeometric function of appell we had to deal with decomposition formulas formulas of differentiation and some adjacent relations for this hypergeometric function in order to get explicit solution of the formulated problem

merit functions for semi definite complementarity problems merit functions such as the gap function the regularized gap function the implicit lagrangian and the norm squared of the fischer burmeister function have played an important role in the solution of complementarity problems defined over the cone of nonnegative real vectors we study the extension of these merit functions to complementarity problems defined over the cone of block diagonal symmetric positive semi definite real matrices the extension suggests new solution methods for the latter problems

an o square root nl iteration large step primal dual affine algorithm for linear programming an algorithm based on reducing a suitable potential function for linear programming is described at each iteration separate scalings are applied to the primal and dual problems and a step is taken in either the primal or the dual space it is shown that a constant reduction can always be achieved leading to a bound of o n 1 2 l iterations moreover it is also shown that a reduction of omega n 1 4 can usually be obtained so that o n 1 4 l iterations are expected to suffice finally it is proved that no general algorithm taking either primal or dual steps and guaranteeing the reduction of such a potential function can achieve r order of convergence greater than one

multiobjective output feedback control via lmi optimization this paper presents an overview of a linear matrix inequality lmi approach to the multiobjective synthesis of linear output feedback controllers the design objectives can be a mix of h sub spl infin performance h sub 2 performance passivity asymptotic disturbance rejection time domain constraints and constraints on the closed loop pole location in addition these objectives can be specified on different channels of the closed loop system when all objectives are formulated in terms of a common lyapunov function controller design amounts to solving a system of linear matrix inequalities the validity of this approach is illustrated by a realistic design example

optimal estimation of deterioration from diagnostic image sequence estimation of mechanical structure damage can greatly benefit from the knowledge that the damage accumulates irreversibly over time this paper formulates a problem of estimation of a pixel wise monotonic increasing or decreasing time series of images from noisy blurred image data our formulation includes temporal monotonicity constraints and a spatial regularization penalty we cast the estimation problem as a large scale quadratic programming qp optimization and describe an efficient interior point method for solving this problem the method exploits the special structure of the qp and scales well to problems with more than a million of decision variables and constraints the proposed estimation approach performs well for simulated data we demonstrate an application of the approach to diagnostic images obtained in structural health monitoring experiments and show that it provides a good estimate of the damage accumulation trend while suppressing spatial and temporal noises

existence and uniqueness of search directions in interior point algorithms for the sdp and the monotone sdlcp various search directions used in interior point algorithms for the semidefinite program sdp and the monotone semidefinite linear complementarity problem sdlcp are characterized by the intersection of a maximal monotone affine subspace and a maximal and strictly antitone affine subspace this observation provides a unified geometric view over the existence of those search directions

paper low order control design for lmi problems using alternating projection methods computational techniques based on alternating projections are proposed to solve control design problems described by linear matrix inequalities lmis in particular we concentrate on the stabilization and the suboptimal h output feedback control design problems these problems can be described by a pair of lmis and an additional coupling condition this coupling condition is convex for the full order control design problem but convexity is lost for the control problem of order strictly less than the plant order we formulate these problems as feasibility problems with matrix constraint sets of simple geometry and we utilize this geometry to obtain analytical expressions for the orthogonal projection operators onto these sets full order and low order controllers are designed using alternating projection methods for the full order controller case global convergence of the alternating projection methods to a feasible solution is guaranteed however for the low order control case only local convergence is guaranteed an example is provided to illustrate the use of these methods for the full order and the low order controller design

polynomial convergence of a new family of primal dual algorithms for semidefinite programming this paper establishes the polynomial convergence of a new class of primal dual interior point path following feasible algorithms for semidefinite programming sdp whose search directions are obtained by applying newton s method to the symmetric central path equation p x p t 1 2 p t s p 1 p x p t 1 2 mu i 0 where p is a nonsingular matrix specifically we show that the short step path following algorithm based on the frobenius norm neighborhood and the semilong step path following algorithm based on the operator 2 norm neighborhood have o sqrt n l and o nl iteration complexity bounds respectively when p i this yields the first polynomially convergent semilong step algorithm based on a pure newton direction restricting the scaling matrix p at each iteration to a certain subset of nonsingular matrices we are able to establish an o n3 2l iteration complexity for the long step path following method the resulting subclass of search directions contains both the nesterov todd direction and the helmberg rendl vanderbei wolkowicz kojima shindoh hara monteiro direction

an implementation of karmarkar s algorithm for linear programming this paper describes the implementation of power series dual affine scaling variants of karmarkar s algorithm for linear programming based on a continuous version of karmarkar s algorithm two variants resulting from first and second order approximations of the continuous trajectory are implemented and tested linear programs are expressed in an inequality form which allows for the inexact computation of the algorithm s direction of improvement resulting in a significant computational advantage implementation issues particular to this family of algorithms such as treatment of dense columns are discussed the code is tested on several standard linear programming problems and compares favorably with the simplex codeminos 4 0

investigations of the properties motion for essential nonlinear systems perturbed by impulses on some hypersurfaces abstract in this work sufficiently conditions boundedness stability asymptotical stability exponential stability attraction practical stability nondisturbed motion systems with impulse perturbations on some hypersurfaces are obtained we formulate the conditions which guarantee that no beating motion of hypersurfaces will be present estimates of cauchy matrix system of variation for approximated system are obtained for different kinds of right side shorted system these estimates are used as one parametric scale increasing functions moreover we also consider two parametric scale such as proposed by demidovich

a potential function reduction algorithm for solving a linear program directly from an infeasible warm start this paper develops an algorithm for solving a standard form linear program directly from an infeasible warm start i e directly from a given infeasible solution hat x that satisfies a hat x b but hat x ngeqslant 0 the algorithm is a potential function reduction algorithm but the potential function is somewhat different than other interior point method potential functions and is given by f x b q ln c t x b sum limits j 1 n ln x j h j c t x b where q n sqrt n is a given constant h is a given strictly positive shift vector used to shift the nonnegativity constaints andb is a lower bound on the optimal value of the linear program the duality gapc t x b is used both in the leading term as well as in the barrier term to help shift the nonnegativity constraints the algorithm is shown under suitable conditions to achieve a constant decrease in the potential function and so achieves a constant decrease in the duality gap and hence also in the infeasibility in o n iterations under more restrictive assumptions regarding the dual feasible region this algorithm is modified by the addition of a dual barrier term and will achieve a constant decrease in the duality gap and in the infeasibility in o sqrt n iterations

linear matrix inequalities riccati equations and indefinite stochastic linear quadratic controls this paper deals with an optimal stochastic linear quadratic lq control problem in infinite time horizon where the diffusion term in dynamics depends on both the state and the control variables in contrast to the deterministic case we allow the control and state weighting matrices in the cost functional to be indefinite this leads to an indefinite lq problem which may still be well posed due to the deep nature of uncertainty involved the problem gives rise to a stochastic algebraic riccati equation sare which is however fundamentally different from the classical algebraic riccati equation as a result of the indefinite nature of the lq problem to analyze the sare we introduce linear matrix inequalities lmis whose feasibility is shown to be equivalent to the solvability of the sare moreover we develop a computational approach to the sare via a semi definite programming associated with the lmis finally numerical experiments are reported to illustrate the proposed approach

the u lagrangian of the maximum eigenvalue function in this paper we apply the u lagrangian theory to the maximum eigenvalue function la and to its precomposition with affine matrix valued mappings we first give geometrical interpretations of the u objects that we introduce we also show that the u lagrangian of la has a hessian which can be explicitly computed the second order development of the u lagrangian provides a second order development of la along a characteristic smooth manifold the set of symmetric matrices whose maximal eigenvalues have a fixed multiplicity the same results can be obtained when we precompose la with an affine matrix valued mapping a provided that this mapping satisfies a regularity condition em transversality condition we show that the hessian of the u lagrangian of la circ a coincides with the reduced hessian encountered in sequential quadratic programming finally we use the u lagrangian to derive second order algorithms for minimizing la circ a

polynomiality of primal dual algorithms for semidefinite linear complementarity problems based on the kojima shindoh hara family of directions kojima shindoh and hara proposed a family of search directions for the semidefinite linear complementarity problem sdlcp and established polynomial convergence of a feasible short step path following algorithm based on a particular direction of their family the question of whether polynomiality could be established for any direction of their family thus remained an open problem this paper answers this question in the affirmative by establishing the polynomiality of primal dual interior point algorithms for sdlcp based on any direction of the kojima shindoh and hara family of search directions we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima et al and monteiro and adler and the predictor corrector algorithm of mizuno et al carry over to the context of sdlcp

the diversity order of the semidefinite relaxation detector in this paper we consider the detection of binary antipodal signals transmitted in a spatially multiplexed fashion over a fading multiple input multiple output mimo channel and where the detection is done by means of semidefinite relaxation sdr the sdr detector is an attractive alternative to maximum likelihood ml detection since the complexity is polynomial rather than exponential assuming that the channel matrix is drawn with independent identically distributed i i d real valued gaussian entries we study the receiver diversity and prove that the sdr detector achieves the maximum possible diversity thus the error probability of the receiver tends to zero at the same rate as the optimal ml receiver in the high signal to noise ratio snr limit this significantly strengthens previous performance guarantees available for the semidefinite relaxation detector additionally it proves that full diversity detection is also possible in certain scenarios when using a noncombinatorial receiver structure

primal dual affine scaling algorithms fail for semidefinite programming in this paper we give an example of a semidefinite programming problem in which primal dual affine scaling algorithms using the hrvw ksh m mt and aho directions fail we prove that each of these algorithms can generate a sequence converging to a non optimal solution and that for the aho direction even its associated continuous trajectory can converge to a non optimal point in contrast with these directions we show that the primal dual affine scaling algorithm using the nt direction for the same semidefinite programming problem always generates a sequence converging to the optimal solution both primal and dual problems have interior feasible solutions and unique optimal solutions which satisfy strict complementarity and are nondegenerate everywhere

an inexact interior point method for system analysis in this article a primal dual interior point algorithm for semidefinite programming that can be used for analysing e g polytopic linear differential inclusions is tailored in order to be more computationally efficient the key to the speedup is to allow for inexact search directions in the interior point algorithm these are obtained by aborting an iterative solver for computing the search directions prior to convergence a convergence proof for the algorithm is given two different preconditioners for the iterative solver are proposed the speedup is in many cases more than an order of magnitude moreover the proposed algorithm can be used to analyse much larger problems as compared to what is possible with off the shelf interior point solvers

a spectral bundle method for semidefinite programming a central drawback of primal dual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices this restricts applicability to problems of small dimension typically semidefinite relaxations arising in combinatorial applications have sparse and well structured cost and coefficient matrices of huge order we present a method that allows us to compute acceptable approximations to the optimal solution of large problems within reasonable time semidefinite programming problems with constant trace on the primal feasible set are equivalent to eigenvalue optimization problems these are convex nonsmooth programming problems and can be solved by bundle methods we propose replacing the traditional polyhedral cutting plane model constructed from subgradient information by a semidefinite model that is tailored to eigenvalue problems convergence follows from the traditional approach but a proof is included for completeness we present numerical examples demonstrating the efficiency of the approach on combinatorial examples

robust performance of linear parametrically varying systems using parametrically dependent linear feedback abstract in this paper a parameter dependent control problem for linear parametrically varying lpv systems is presented sufficient conditions are given that guarantee an lpv system is exponentially stable and achieves an induced l 2 norm performance objective from the disturbance to error signals the usefulness of parameter dependent controllers is motivated from a gain scheduling viewpoint the resulting synthesis problem is reformulated into a convex optimization problem which can be solved using efficient new algorithms

interior point trajectories in semidefinite programming in this paper we study interior point trajectories in semidefinite programming sdp including the central path of an sdp this work was inspired by the seminal work of megiddo on linear programming trajectories progress in math programming interior point algorithms and related methods n megiddo ed springer verlag berlin 1989 pp 131 158 under an assumption of primal and dual strict feasibility we show that the primal and dual central paths exist and converge to the analytic centers of the optimal faces of respectively the primal and the dual problems we consider a class of trajectories that are similar to the central path but can be constructed to pass through any given interior feasible or infeasible point and study their convergence finally we study the derivatives of these trajectories and their convergence

a one parametric class of merit functions for the symmetric cone complementarity problem abstract in this paper we extend the one parametric class of merit functions proposed by kanzow and kleinmichel c kanzow h kleinmichel a new class of semismooth newton type methods for nonlinear complementarity problems comput optim appl 11 1998 227 251 for the nonnegative orthant complementarity problem to the general symmetric cone complementarity problem sccp we show that the class of merit functions is continuously differentiable everywhere and has a globally lipschitz continuous gradient mapping from this we particularly obtain the smoothness of the fischer burmeister merit function associated with symmetric cones and the lipschitz continuity of its gradient in addition we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the ncp function classes studied by c kanzow y yamashita m fukushima new ncp functions and their properties j optim theory appl 97 1997 115 135 to the sccp by exploiting the cartesian p properties for a nonlinear transformation we show that the class of regularized merit functions provides a global error bound for the solution of the sccp and moreover has bounded level sets under a rather weak condition which can be satisfied by the monotone sccp with a strictly feasible point or the sccp with the joint cartesian r 02 property all of these results generalize some recent important works in j s chen p tseng an unconstrained smooth minimization reformulation of the second order cone complementarity problem math program 104 2005 293 327 c k sim j sun d ralph a note on the lipschitz continuity of the gradient of the squared norm of the matrix valued fischer burmeister function math program 107 2006 547 553 p tseng merit function for semidefinite complementarity problems math program 83 1998 159 185 under a unified framework

a self correcting version of karmarkar s algorithm a relaxed version of karmarkar s algorithm is developed that does not require the direction of movement to be in the null space of the constraint matrix the algorithm is proved to have the same rate of convergence as karmarkar s algorithm conditions are also given guaranteeing that the approach suggested by karmarkar for transforming an inequality form linear program into the form required by his algorithm is well defined

robust pole placement in lmi regions discusses analysis and synthesis techniques for robust pole placement in linear matrix inequality lmi regions a class of convex regions of the complex plane that embraces most practically useful stability regions the focus is on linear systems with static uncertainty on the state matrix for this class of uncertain systems the notion of quadratic stability and the related robustness analysis tests are generalized to arbitrary lmi regions the resulting tests for robust pole clustering are all numerically tractable because they involve solving linear matrix inequalities lmis and cover both unstructured and parameter uncertainty these analysis results are then applied to the synthesis of dynamic output feedback controllers that robustly assign the closed loop poles in a prescribed lmi region with some conservatism this problem is again tractable via lmi optimization in addition robust pole placement can be combined with other control objectives such as h sub 2 or h sub spl infin performance to capture realistic sets of design specifications physically motivated examples demonstrate the effectiveness of this robust pole clustering technique

centers of monotone generalized complementarity problems let c be a full dimensional closed pointed and convex cone in a finite dimensional real vector space e with an inner product of x y e and m a maximal monotone subset of e e this paper studies the existence and continuity of centers of the monotone generalized complementarity problem associated with c and m find x y m c c such that 0 here c y e 0 for all x c denotes the dual cone of c the main result of the paper unifies and extends some results established for monotone complementarity problems in euclidean space and monotone semidefinite linear complementarity problems in symmetric matrices

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

a large step analytic center method for a class of smooth convex programming problems in this paper a large step analytic center method for smooth convex programming is proposed the method is a natural implementation of the classical method of centers it is assumed that the objective and constraint functions fulfil the so called relative lipschitz condition with lipschitz constant m 0 a great advantage of the method over the existing path following methods is that the steps can be made long by performing linesearches in this method linesearches are performed along the newton direction with respect to a strictly convex potential function when located far away from the central path when sufficiently close to this path a lower bound for the optimal value is updated it is proven that the number of iterations required by the algorithm to converge to an epsilon optimal solution is o 1 m 2 sqrt n ln epsilon or o 1 m 2 n ln epsilon depending on the updating scheme for the lower bound

a unified approach to lmi based reduced order self scheduling control synthesis abstract we consider a reduced order controller synthesis for a general class of control specifications for linear parameter varying lpv systems when some of state variables are exactly available the class is defined in an abstract manner so that it uniformly deals with many significant specifications a necessary and sufficient condition for the existence of a reduced order controller is given in terms of linear matrix inequalities lmis we also show that the order of the controller can be reduced by the number of the state variables exactly available in the measurements moreover in the case of linear time invariant lti systems a parameterization of all desirable reduced order lti controllers is given by means of solutions of lmis the results in this paper generalize the class of control specifications in which a reduced order controller exists making it possible to synthesize a reduced order controller based on lmis for multi objective control specifications furthermore these results uniformly describe and generalize the existing results on synthesis of a constant state and a full order output feedback controller for lti and lpv systems such that the specification is given by the existence of a constant positive definite matrix

a recurrent neural network for real time semidefinite programming the semidefinite programming problem is an important optimization problem that has been extensively investigated a real time solution method for solving such a problem however is still not yet available the paper proposes a recurrent neural network for this purpose first an auxiliary cost function is introduced to minimize the duality gap between the admissible points of the primal problem and the corresponding dual problem then a dynamical system is constructed to drive the duality gap to zero exponentially along any trajectory by modifying the gradient of the auxiliary cost function furthermore a subsystem is developed to circumvent the computation of matrix inverse so that the resulting overall dynamical system can be realized using a recurrent neural network the architecture of the resulting neural network is discussed the operating characteristics and performance of the proposed approach are demonstrated by means of simulation results

affine scaling algorithm fails for semidefinite programming in this paper we introduce an affine scaling algorithm for semidefinite programming sdp and give an example of a semidefinite program such that the affine scaling algorithm converges to a non optimal point both our program and its dual have interior feasible solutions and unique optimal solutions which satisfy strict complementarity and they are non degenerate everywhere

on quasi stability for impulsive differential systems on developpe une theorie de la stabilite relativement a une solution donnee de systemes differentiels impulsifs une telle etude conduit a un mecanisme complique et les notions de stabilite doivent etre modifiees

the simplest semidefinite programs are trivial we consider optimization problems of the following type min left mbox tr cx colon ax b x quad mbox positive semidefinite right here tr denotes the trace operator c and x are symmetric n n matrices b is a symmetric m m matrix and a denotes a linear operator such problems are called semidefinite programs and have recently become the object of considerable interest due to important connections with max min eigenvalue problems and with new bounds for integer programming in the context of symmetric matrices the simplest linear operators have the following form ax mxm t where m is an arbitrary m n matrix in this paper we show that for such linear operators the optimization problem is trivial in the sense that an explicit solution can be given

a primal dual potential reduction method for problems involving matrix inequalities we describe a potential reduction method for convex optimization problems involving matrix inequalities the method is based on the theory developed by nesterov and nemirovsky and generalizes gonzaga and todd s method for linear programming a worst case analysis shows that the number of iterations grows as the square root of the problem size but in practice it appears to grow more slowly as in other interior point methods the overall computational effort is therefore dominated by the least squares system that taust be solved in each iteration a type of conjugate gradient algorithm can be used for this purpose which results in important savings for two reasons first it allows us to take advantage of the special structure the problems often have e g lyapunov or algebraic riccati inequalities second we show that the polynomial bound on the number ofiterations remains valid even if the conjugate gradient algorithm is not run until completion which in practice can greatly reduce the computational effort per iteration we describe in detail how the algorithm works for optimization problems with l lyapunov inequaliues each of size m we prove an overall worst case operation count of o mssll5 the average case complexity appears to be closer to o m4ll5 this estimate is justified by extensive numerical experimentation and is consistent with other researchers experience with the practical performance of interior point algorithms for linear programming this result means that the computational cost of extending current control theory based on the solution of lyapunov or riccati equations to a theory that is based on the solution of multiple coupled lyapunov or riccati inequalities is modest

a relaxed revision of karmarkar s method a relaxed version of karmarkar s method is developed this method is proved to have the same polynomial time complexity as karmarkar s method and its efficient implementation using inexact projections is discussed computational results obtained using a preliminary implementation of the method are presented which indicate that the method is practicable

the bounded smooth reformulation and a trust region algorithm for semidefinite complementarity problems in this paper we present a new merit function for the semidefinite complementarity problem sdcp by extending the bounded smooth reformulation for variational inequality problems we prove that the merit function has a bounded level set and any stationary point of it is a global minimizer without the assumption of monotonicity moreover we present a trust region algorithm for solving the minimization problem with semidefinite constraints the trust region subproblem is solved by the truncated conjugate gradient method and the global convergence is established even without requiring the existence of an accumulation point of the generated sequence

method of centers for minimizing generalized eigenvalues abstract we consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices each of which depends affinely on the decision variables although this problem may appear specialized it is in fact quite general and includes for example all linear quadratic and linear fractional programs many problems arising in control theory can be cast in this form the problem is nondifferentiable but quasiconvex so methods such as kelley s cutting plane algorithm or the ellipsoid algorithm of shor nemirovsky and yudin are guaranteed to minimize it in this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently the algorithm is a variation on huard s method of centers using a self concordant barrier for matrix inequalities developed by nesterov and nemirovsky nesterov and nemirovsky have also extended their potential reduction methods to handle the same problem since the problem is quasiconvex but not convex devising a nonheuristic stopping criterion i e one that guarantees a given accuracy is more difficult than in the convex case we describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper in some cases than a more general result to nesterov and nemirovsky the algorithm is demonstrated on an example determining the quadratic lyapunov function that optimizes a decay rate estimate for a differential inclusion

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

explicit controller formulas for lmi based h synthesis the set of h controllers with closed loop performance c can be implicitly parametrized by the solutions r s of a system of linear matrix inequalities lmi the matrices r and s play a role analogous to that of the riccati solutions x and y in classical riccati based h control useful applications include lmi based h synthesis mixed h 2h design and h design with a pole placement constraint this paper is concerned with the reliable computation of h controllers given a solution r s of the characteristic system of lmis explicit formulas are derived for both the regular and singular cases remarkably these formulas are extensions of the usual central controller formulas where the lmi solutions r and s replace the riccati solutions x and y simple and numerically appealing new formulas for discrete time h controllers are also derived

large scale optimization of eigenvalues optimization problems involving eigenvalues arise in many applications let x be a vector of real parameters and let a x be a continuously differentiable symmetric matrix function of x we consi

robustness under bounded uncertainty with phase information the authors consider uncertain linear systems where the uncertainties in addition to being bounded also satisfy constraints on their phase in this context the authors define the phase sensitive structured singular value ps ssv of a matrix and show that sufficient and sometimes necessary conditions for stability of such uncertain linear systems can be rewritten as conditions involving ps ssv they then derive upper bounds for ps ssv computable via convex optimization they extend these results to the case where the uncertainties are structured diagonal or block diagonal for instance

an interior point method for minimizing the maximum eigenvalue of a linear combination of matrices we consider the problem of finding lambda opt mathop inf limits x in r m lambda max a 0 sum limits i 1 m x i a i 1 and an optimal solution xopt if it exists here the a i are given symmetric matrices in ir n n for 0 i m and max a is the largest eigenvalue of a matrix a problem 1 can be rewritten as a convex differentiable problem with a positive definite constraint positive definiteness can easily be verified numerically via cholesky decomposition moreover the cone of positive definite matrices a is characterized by the domain of the smooth convex barrier function log det a

integro sum inequalities and motion stability of systems with impulse perturbations abstract this article is devoted to the investigation of the properties of motion represented by essential nonlinear system of differential equations with impulse effect the results based on the solvability problem by s a chaplygin for discontinuous functions satisfy integro sum inequalities the conditions of boundedness stability asymptotical stability practical stability attraction motion for different kinds of nonlinearities of systems are obtained

an ol n 3 potential reduction algorithm for linear programming we describe a primal dual potential function for linear programming phi x s rho ln x t s sum limits j 1 n ln x j s j where n x is the primal variable ands is the dual slack variable as a result we develop an interior point algorithm seeking reductions in the potential function with rho n sqrt n neither tracing the central path nor using the projective transformation the algorithm converges to the optimal solution set in o sqrt n l iterations and uses o n3l total arithmetic operations we also suggest a practical approach to implementing the algorithm

on gronwall bellman bihari type integral inequalities in several variables with retardation the celebrated gronwall bellman bihari inequality and its various linear and nonlinear generalizations have been of vital importance in the study of existence uniqueness continuous dependence comparison perturbation boundedness and stability results of ordinar differential equations and integral equations 2 6 14 15 18 20 22 23 371 in recent years the more than one variable generalizations of this inequality have evoked considerable interest since they are handy tools in the theory of partial differential and integral equations specifically we refer to the results of bondge and pachpatte 7 11 131 bondge et 121 chu and metcalf 161 colan and diaz 171 headley 21 agarwal 11 pachpatte 24 301 rasmussen 31 shih and yeh 32 41 singare and pachpatte 33 snow 34 35 than dapani and agarwal 36 yeh 39 40 42 young 43 and zahariev and bainov 44 among others in this paper we establish some new nonlinear integral inequalities with delay in n independent variables which include and unify recent results on integral inequalities in several variables by shih and yeh 41 yeh 42 and zahariev and bainov 44 we also deduce from our results new generalizations of earlier results of bihari 6 bellman 4 dhongade and deo 18 191 and pachpatte 22 231 some applications of our results to a certain class of nonlinear hyperbolic functional partial differential equations are also given

projection and contraction methods for semidefinite programming recently he 1992 1994a b 1996 presented some iterative projection and contraction pc methods for linear variational inequalities its simplicity robustness and ability to handle large problems has attracted a lot of attention this paper extends the pc methods from linear programming to semidefinite programming sdp the sdp problem is transformed into an equivalent projection equation which is solved by pc methods some promising numerical results obtained from a preliminary implementation are included

brief lpv systems with parameter varying time delays analysis and control in this paper we address the analysis and state feedback synthesis problems for linear parameter varying lpv systems with parameter varying time delays it is assumed that the state space data and the time delays are dependent on parameters that are measurable in real time and vary in a compact set with bounded variation rates we explore the stability and the induced l 2 norm performance of these systems using parameter dependent lyapunov functionals in addition the design of parameter dependent state feedback controllers that guarantee desired l 2 gain performance is examined both analysis and synthesis conditions are formulated in terms of linear matrix inequalities lmis that can be solved via efficient interior point algorithms

semi definite matrix constraints in optimization positive semi definite matrix constraints arise in a number of optimization problems in which some or all of the elements of a matrix are variables such as the educational testing and matrix modification problems the structure of such constraints is developed including expressions for the normal cone feasible directions and their application to optimality conditions a computational framework is given within which these concepts can be exploited and which permits the quantification of second order effects the matrix of lagrange multipliers in this formulation is shown to have an important relationship to the characterization of the normal cone modifications of various iterative schemes in nonlinear programming are considered in order to develop an effective algorithm for the educational testing problem and comparative numerical experiments are described it is shown that a particular choice of the penalty parameter for an l 1 exact penalty function is appropriate for this type of problem the be

explicit solutions for interval semidefinite linear programs abstract we consider the special class of semidefinite linear programs ivp maximize trace cx subject to l a x u where c x l u are symmetric matrices a is an onto linear operator and denotes the lowner positive semidefinite partial order we present explicit representations for the general primal and dual optimal solutions this extends the results for standard linear programming that appeared in ben israel and charnes 3 this work is further motivated by the explicit solutions for a different class of semidefinite problems presented recently in yang and vanderbei 15

a primal dual interior point method for robust optimal control of linear discrete time systems this paper describes how to efficiently solve a robust optimal control problem using recently developed primal dual interior point methods one potential application is model predictive control the optimization problem considered consists of a worst case quadratic performance criterion over a finite set of linear discrete time models subject to inequality constraints on the states and control signals the scheme has been prototyped in matlab to give a rough idea of the efficiencies obtained it is possible to solve problems with more than 10 000 primal variables and 40 000 constraints on a workstation the key to the efficient implementation is an iterative solver in conjunction with a riccati recursion invertible pre conditioner for computing the search directions

some new integral inequalities of bellman bihari type with delay for discontinuous functions abstract in this article we investigate some integral functional inequalities of bellman bihari type for piecewise continuous functions with some fixed points of discontinuity we also prove a new analogy and generalization of results which were obtained by bellman and bihari to integro sum inequalities with delay and discontinuities that do not belong to lipschitz s type

nl sub q theory checking and imposing stability of recurrent neural networks for nonlinear modeling it is known that many discrete time recurrent neural networks such as e g neural state space models multilayer hopfield networks and locally recurrent globally feedforward neural networks can be represented as nl sub q systems sufficient conditions for global asymptotic stability and input output stability of nl sub q systems are available including three types of criteria 1 diagonal scaling 2 criteria depending on diagonal dominance 3 condition number factors of certain matrices the paper discusses how narendra s 1990 1991 dynamic backpropagation procedure which is used for identifying recurrent neural networks from i o measurements can be modified with an nl sub q stability constraint in order to ensure globally asymptotically stable identified models an example illustrates how system identification of an internally stable model corrupted by process noise may lead to unwanted limit cycle behavior and how this problem can be avoided by adding the stability constraint

duality and basis functions for robust h 2 performance analysis the robust h2 performance of a plant with a prescribed uncertain component refers to its white noise rejection level in the worst case with respect to its uncertainty set in this paper we derive a dual formulation for the computation of the robust h2 performance of a plant with unit gain structured uncertainty the dual framework gives a tight lower bound on robust h2 performance with no duality gap with respect to the primal approach paganini 1995a we also show how the upper and lower bounds for the robust h2 performance of a plant may be computed using semidefinite programming linear matrix inequality type formulae based on basis functions and provide an example which enables us to demonstrate the relative merits and the complementary nature of the basis function and the gridding approaches to robust h2 performance analysis for both the primal and the dual formulations