almost invariant subspaces an approach to high gain feedback design part ii almost conditionally invariant subspaces in part i of this paper almost controlled invariant subspaces were studied in this part we will consider their duals the almost conditionally invariant subspaces these concepts give immediately by dualization of the almost disturbance decoupling control by state feedback the solution of the almost disturbance decoupled estimation problem finally we consider the problem of approximate disturbance decoupling by measurement feedback and it is shown that this problem is solvable to any arbitrary degree of accuracy if and only if 1 almost disturbance decoupling by state feedback and 2 almost disturbance decoupled estimation of the to be controlled output are both possible

on optimality conditions for structured families of nonlinear programming problems optimality conditions for families of nonlinear programming problems inrn are studied from a generic point of view the objective function and some of the constraints are assumed to depend on a parameter while others are held fixed techniques of differential topology are used to show that under suitable conditions certain strong second order conditions are necessary for optimality except possibly for parameter values lying in a negligible set

on the generic properties of convex optimization problems in conic form we prove that strict complementarity primal and dual nondegeneracy of optimal solutions of convex optimization problems in conic form are generic properties in this paper we say generic to mean that the set of data possessing the desired property or properties has strictly larger hausdorff dimension than the set of data that does not our proof is elementary and it employs an important result due to larman 7 on the boundary structure of convex bodies

practical aspects of the moreau yosida regularization theoretical preliminaries when computing the infimal convolution of a convex function f with the squared norm the so called moreau yosida regularization of f is obtained among other things this function has a lipschitzian gradient we investigate some more of its properties relevant for optimization the most important part of our study concerns second order differentiability existence of a second order development of f implies that its regularization has a hessian for the converse we disclose the importance of the decomposition of bbb r n along cal u the subspace where f is smooth and cal v the subspace parallel to the subdifferential of f

identifying structure of nonsmooth convex functions by the bundle technique we consider the problem of minimizing nonsmooth convex functions defined piecewise by a finite number of functions each of which is either convex quadratic or twice continuously differentiable with positive definite hessian on the set of interest this is a particular case of functions with primal dual gradient structure a notion closely related to the so called cal vu space decomposition at a given point nonsmoothness is locally restricted to the directions of the subspace cal v while along the subspace cal u the behavior of the function is twice differentiable constructive identification of the two subspaces is important because it opens the way to devising fast algorithms for nonsmooth optimization by following iteratively the manifold of smoothness on which superlinear cal u newton steps can be computed in this work we show that for the class of functions in consideration the information needed for this identification can be obtained from the output of a standard bundle method for computing proximal points provided a minimizer satisfies the nondegeneracy and strong transversality conditions

structured systems within the transfer matrix approach application to decoupling abstract we consider a new approach to structured systems within the transfer matrix framework we associate with such a structured system a graph which concentrates both knowledge on structural properties of independent subsystems and on their interconnections the analysis of this associated graph allows to conclude on the decouplability of structured systems

on eigenvalue optimization in this paper we study optimization problems involving eigenvalues of symmetric matrices one of the difficulties with numerical analysis of such problems is that the eigenvalues considered as functions of a symmetric matrix are not differentiable at those points where they coalesce we present a general framework for a smooth differentiable approach to such problems it is based on the concept of transversality borrowed from differential geometry in that framework we discuss first and second order optimality conditions and rates of convergence of the corresponding second order algorithms finally we present some results on the sensitivity analysis of such problems

a algorithm for convex minimization for convex minimization we introduce an algorithm based on space decomposition the method uses a bundle subroutine to generate a sequence of approximate proximal points when a primal dual track leading to a solution and zero subgradient pair exists these points approximate the primal track points and give the algorithm s or corrector steps the subroutine also approximates dual track points that are gradients needed for the method s newton predictor steps with the inclusion of a simple line search the resulting algorithm is proved to be globally convergent the convergence is superlinear if the primal dual track points and the objective s hessian are approximated well enough

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

semidefinite representation of convex sets let s x in mathbb r n g 1 x geq 0 ldots g m x geq 0 be a semialgebraic set defined by multivariate polynomials g i x assume s is convex compact and has nonempty interior let s i x in mathbb r n g i x geq 0 and s resp s i be the boundary of s resp s i this paper as does the subject of semidefinite programming sdp concerns linear matrix inequalities lmis the set s is said to have an lmi representation if it equals the set of solutions to some lmi and it is known that some convex s may not be lmi representable helton and vinnikov in commun pure appl math 60 5 654 674 2007 a question arising from nesterov and nemirovski siam studies in applied mathematics society for industrial and applied mathematics siam philadelphia 1994 see helton and vinnikov in commun pure appl math 60 5 654 674 2007 and nemirovski in plenary lecture international congress of mathematicians icm madrid spain 2006 is given a subset s of mathbb r n does there exist an lmi representable set in some higher dimensional space mathbb r n n whose projection down onto mathbb r n equals s such s is called semidefinite representable or sdp representable this paper addresses the sdp representability problem the following are the main contributions of this paper i assume g i x are all concave on s if the positive definite lagrange hessian condition holds i e the hessian of the lagrange function for optimization problem of minimizing any nonzero linear function l t x on s is positive definite at the minimizer then s is sdp representable ii if each g i x is either sos concave 2 g i x w x t w x for some possibly nonsquare matrix polynomial w x or strictly quasi concave on s then s is sdp representable iii if each s i is either sos convex or poscurv convex s i is compact convex whose boundary has positive curvature and is nonsingular i e g i x 0 on s i s then s is sdp representable this also holds for s i for which s i s extends smoothly to the boundary of a poscurv convex set containing s iv we give the complexity of schmudgen and putinar s matrix positivstellensatz which are critical to the proofs of i iii

genericity results in linear conic programming a tour d horizon this paper is concerned with so called generic properties of general linear conic programs many results have been obtained on this subject during the last two decades for example it is known that uniqueness strict complementarity and nondegeneracy of optimal solutions hold for almost all problem instances strong duality holds generically in a stronger sense i e it holds for a generic subset of problem instances in this paper we survey known results and present new ones in particular we give an easy proof of the fact that slater s condition holds generically in linear conic programming we further discuss the problem of stability of uniqueness nondegeneracy and strict complementarity we also comment on the fact that in general a conic program cannot be treated as a smooth problem and that techniques from nonsmooth geometric measure theory are needed

active sets nonsmoothness and sensitivity nonsmoothness pervades optimization but the way it typically arises is highly structured nonsmooth behavior of an objective function is usually associated locally with an active manifold on this manifold the function is smooth whereas in normal directions it is vee shaped active set ideas in optimization depend heavily on this structure important examples of such functions include the pointwise maximum of some smooth functions and the maximum eigenvalue of a parametrized symmetric matrix among possible foundations for practical nonsmooth optimization this broad class of partly smooth functions seems a promising candidate enjoying a powerful calculus and sensitivity theory in particular we show under a natural regularity condition that critical points of partly smooth functions are stable small perturbations to the function cause small movements of the critical point on the active manifold

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

a proximal method for composite minimization we consider minimization of functions that are compositions of convex or prox regular functions possibly extended valued with smooth vector functions a wide variety of important optimization problems fall into this framework we describe an algorithmic framework based on a subproblem constructed from a linearized approximation to the objective and a regularization term properties of local solutions of this subproblem underlie both a global convergence result and an identification property of the active manifold containing the solution of the original problem preliminary computational results on both convex and nonconvex examples are promising

survey generic properties and control of linear structured systems a survey in this survey paper we consider linear structured systems in state space form where a linear system is structured when each entry of its matrices like a b c and d is either a fixed zero or a free parameter the location of the fixed zeros in these matrices constitutes the structure of the system indeed a lot of man made physical systems which admit a linear model are structured a structured system is representative of a class of linear systems in the usual sense it is of interest to investigate properties of structured systems which are true for almost any value of the free parameters therefore also called generic properties interestingly a lot of classical properties of linear systems can be studied in terms of genericity moreover these generic properties can in general be checked by means of directed graphs that can be associated to a structured system in a natural way we review here a number of results concerning generic properties of structured systems expressed in graph theoretic terms by properties we mean here system specific properties like controllability the finite and infinite zero structure and so on as well as solvability issues of certain classical control problems like disturbance rejection input output decoupling and so on in this paper we do not try to be exhaustive but instead by a selection of results we would like to motivate the reader to appreciate what we consider as a wonderful modelling and analysis tool we emphasize the fact that this modelling technique allows us to get a number of important results based on poor information on the system only moreover the graph theoretic conditions are intuitive and are easy to check by hand for small systems and by means of well known polynomially bounded combinatorial techniques for larger systems

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

coset codes for partial response channels or coset codes with spectral nulls known coset codes are adapted for use on partial response channels or to generate signals with spectral nulls by using coset precoding and running digital sum feedback any desired tradeoff can be achieved between the power and spectra of the relevant sequences up to the optimum tradeoff possible a fundamental theorem specifying this optimum tradeoff is given a maximum likelihood sequence estimation mlse decoder for the original code may be used for the adapted code and such a decoder then attains the minimum squared distance of the original code these methods sometimes generate codes with greater minimum squared distance than that of the original code this distance can be attained by augmented decoders although such decoders inherently require long decoding delays and may be subjected to quasi catastrophic error propagation the authors conclude that at least for sequences supporting large numbers of bits per symbol coset codes can be adapted to achieve effectively the same performance and complexity on partial response channels or for sequences with spectral nulls as they do in the ordinary memoryless case

on the structure at infinity of a structured system abstract we develop a graph theoretic characterization of the generic structure at infinity of the transfer matrix of a structured system we show that the generic structure at infinity can be determined by means of algorithms from combinatorial optimization based on the max flow min cut theorem and on results concerning minimal cost flows as an application of the obtained characterization we propose a structural version of two well known disturbance decoupling problems and we derive graph theoretic necessary and sufficient conditions for the solvability of each of the two problems

identifiable surfaces in constrained optimization the concept of a class c p identifiable surface of a convex set in euclidean space is introduced the paper shows how the smoothness of these surfaces is related to the smoothness of the projection operator and presents finite identification results for certain algorithms for minimization of a function over this set the work uses a partially geometric view of constrained optimization to generalize previous finite identification results

the structure of completely positive matrices according to their cp rank and cp plus rank we study the topological properties of the cp rank operator mathrm cp a and the related cp plus rank operator mathrm cp a which is introduced in this paper in the set mathcal s n of symmetric n times n matrices for the set of completely positive matrices mathcal cp n we show that for any fixed p the set of matrices a satisfying mathrm cp a mathrm cp a p is open in mathcal s n setminus mathrm bd mathcal cp n we also prove that the set mathcal a n of matrices with mathrm cp a mathrm cp a is dense in mathcal s n by applying the theory of semi algebraic sets we are able to show that membership in mathcal a n is even a generic property we furthermore answer several questions on the existence of matrices satisfying mathrm cp a mathrm cp a or mathrm cp a neq mathrm cp a and establish genericity of having infinitely many minimal cp decompositions

structural invertibility of transfer functions in this note we study certain properties of a transfer function in terms of the structure of the state space representation a b c specifically the rank of the transfer function and the degree of a decoupled system which can be obtained by applying a left decoupler are characterized in terms of the structure of a b c

generic properties for semialgebraic programs in this paper we study genericity for the class of semialgebraic optimization problems with equality and inequality constraints in which every problem of the class is obtained by linear perturbations of the objective function while the core objective function and the constraint functions are kept fixed assume that the linear independence constraint qualification is satisfied at every point in the constraint set it is shown that almost all problems in the class are such that i the restriction of the objective function on the constraint set is coercive and regular at infinity ii there is a unique optimal solution lying on a unique active manifold at which the strict complementarity and second order sufficiency conditions the quadratic growth condition and the holder type global error bound hold and iii all minimizing sequences converge furthermore the active manifold is constant and the optimal solution and the optimal value function depend analytically under local perturbations of the

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

transfer function conditions for c a b pairs state space conditions for the solution of the output feedback disturbance rejection problem have been given in a recent paper 1 in terms of c a b pairs this paper develops constructive and simple solvability conditions for this problem in terms of the rational matrix equation g s x s n s h s

the optimum combination of block codes and receivers for arbitrary channels a determination is made of the combinations of transmitted block codewords and matching receiver filters that maximize the signal to noise ratio at the detector with arbitrary channel impulse response and arbitrary noise spectrum this permits representing the channel as a set of parallel uncoupled subchannels with uncorrelated noise the representation is used to calculate the capacity of an arbitrary block coded time discrete channel it is shown that substantial coding gains compared to symbol by symbol processing can be achieved the block code used can be concatenated with trellis coded modulation so that the two coding gains are additive

trellis coding for partial response channels we consider trellis coding techniques for improving the reliability of digital transmission over noisy partial response channels such channels are commonly encountered in digital communication systems and also play a role in devices for data recording concentrating on the channels with characteristics 1 mp d we study methods to obtain codes which increase free euclidean distance between permitted sequences of channel outputs and avoid the occurrence of unlimited runs of identical outputs at the expense of some loss in data rate one technique employs the concept of set partitioning the other is based on using convolutional codes designed for maximum free hamming distance in conjunction with a precoder both methods lead to essentially equivalent codes

a note on alternating projections for ill posed semidefinite feasibility problems we observe that sturm s error bounds readily imply that for semidefinite feasibility problems the method of alternating projections converges at a rate of mathcal o big k frac 1 2 d 1 2 big o k 12d 1 2 where d is the singularity degree of the problem the minimal number of facial reduction iterations needed to induce slater s condition consequently for almost all such problems in the sense of lebesgue measure alternating projections converge at a worst case rate of mathcal o big frac 1 sqrt k big o 1k

a geometric approach for structured systems application to disturbance decoupling structured systems described by state space models are considered for these systems the entries of the state space model matrices are supposed to be either fixed zeros or free independent parameters for such systems one can study structural properties i e properties that are valid for almost any value of the parameters we revisit the classical geometric theory in the context of structured systems in particular the well known notions of a b invariant and c a invariant subspaces are redefined and analysed in this context we characterize fixed subspaces included or containing these invariant subspaces and show that they play a key role in solving control problems as an application of this geometric approach for structured systems we propose new solutions to the disturbance decoupling problems by state feedback and by output measurement feedback the solvability conditions are easily checked on the associated graph of the system

trellis coded modulation with multidimensional constellations trellis coded modulation schemes using four eight or 16 dimensional constellations have a number of potential advantages over the usual two dimensional schemes a smaller constituent two dimensional constellation easier tolerance to phase ambiguities and a better trade off between complexity and coding gain a number of such schemes are presented and evaluated starting with a variety of multidimensional lattices we show how to select multidimensional constellations how to partition them into subsets how to construct trellis codes using those subsets and how to map hits to constellation points simplifications of the viterbi decoding algorithm are presented we conclude that there are multidimensional trellis coded modulation schemes that perform better for the same complexity than do two dimensional schemes

generic minimizing behavior in semialgebraic optimization we present a theorem of sard type for semialgebraic set valued mappings whose graphs have dimension no larger than that of their range space the inverse of such a mapping admits a single valued analytic localization around any pair in the graph for a generic value parameter this simple result yields a transparent and unified treatment of generic properties of semialgebraic optimization problems typical semialgebraic problems have finitely many critical points around each of which they admit a unique active manifold analogue of an active set in nonlinear optimization moreover such critical points satisfy strict complementarity and second order sufficient conditions for optimality are indeed necessary

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

the dimension of semialgebraic subdifferential graphs a b s t r a c t a corollary of a celebrated theorem of minty is that the subdifferential graph of a closed convex function on r n has uniform local dimension n in contrast there exist nonconvex closed functions whose subdifferentials have large graphs we consider how far minty s corollary extends to functions that are nonconvex but semi algebraic

disturbance decoupling by measurement feedback for structured transfer matrix systems structured transfer matrix systems are linear systems given by transfer matrices of which the infinite zero order of each nonzero entry is known while the associated infinite gains are unknown and assumed mutually independent in this paper necessary and sufficient conditions are derived for the generic solvability of the well known disturbance decoupling problem by measurement feedback for structured transfer matrix systems generic solvability here means solvability for almost all possible values for the infinite gains of the nonzero transfer matrix entries the conditions will be stated in terms of a bipartite graph that can be easily associated with a structured transfer matrix system the advantage of this is that the conditions then can be verified by means of graph theoretic algorithms

new trellis codes based on lattices and cosets a new technique is proposed for constructing trellis codes which provides an alternative to ungerboeck s method of set partitioning the new codes use a signal constellation consisting of points from an n dimensional lattice lambda with an equal number of points from each coset of a sublattice lambda one part of the input stream drives a generalized convolutional code whose outputs are cosets of lambda while the other part selects points from these cosets several of the new codes are better than those previously known

vector coding for partial response channels a linear technique for combining equalization and coset codes on partial response channels with additive white gaussian noise is developed the technique vector coding uses a set of transmit filters or vectors to partition the channel into an independent set of parallel intersymbol interference isi free channels for any given finite or infinite block length the optimal transmit vectors for such channel partitioning are shown to be the eigenvectors of the channel covariance matrix for the specified block length and the gains of the individual channels are the eigenvalues an optimal bit allocation and energy distribution are derived for the set of parallel channels under an accurate extension of the continuous approximation for power in optimal multidimensional signal sets for constellations with unequal signal spacing in different dimensions examples are presented that demonstrate performance advantages with respect to zero forcing decision feedback methods that use the same coset code on the same partial response channel only resampling the channel at an optimal rate and assuming no errors in the feedback path will bring the performance of the decision feedback methods up to the level of the vector coded system

on the identification of active constraints nondegeneracy conditions that guarantee that the optimal active constraints are identified in a finite number of iterations are studied results of this type have only been established for a few algorithms and then under restrictive hypothesis the main result is a characterization of those algorithms that identify the optimal constraints in a finite number of iterations this result is obtained with a nondegeneracy assumption which is equivalent in the standard nonlinear programming problem to the assumption that there is a set of strictly complementary lagrange multipliers as an important consequence of the authors results the way that this characterization applies to gradient projection and sequential quadratic programming algorithms is shown

a frank wolfe type theorem for nondegenerate polynomial programs in this paper we study the existence of optimal solutions to a constrained polynomial optimization problem more precisely let f 0 f 0 and f 1 ldots f p mathbb r n rightarrow mathbb r f 1 f p r n r be convenient polynomial functions and let s x in mathbb r n f i x le 0 i 1 ldots p ne emptyset s x r n f i x 0 i 1 p under the assumption that the map f 0 f 1 ldots f p mathbb r n rightarrow mathbb r p 1 f 0 f 1 f p r n r p 1 is non degenerate at infinity we show that if f 0 f 0 is bounded from below on s s then f 0 f 0 attains its infimum on s s

output feedback disturbance decoupling graph interpretation for structured systems abstract in this paper we consider first the output feedback disturbance decoupling problem for linear systems we derive a set of new necessary or sufficient conditions for the solvability of this problem these conditions are expressed in terms of system structural invariants such as infinite zero orders and essential orders then we consider the output feedback disturbance decoupling problem for structured linear systems necessary or sufficient conditions for the generic solvability of this problem are provided they are expressed in terms of input output path length comparisons on the system associated graph

semi algebraic functions have small subdifferentials we prove that the subdifferential of any semi algebraic extended real valued function on mathbf r n has n dimensional graph we discuss consequences for generic semi algebraic optimization problems

disturbance rejection for structured systems the disturbance rejection problem for structured systems is solved a graph is associated with such a structured system disturbance rejection is shown to be solvable by state feedback whenever the control inputs reach the outputs more quickly than the disturbances in the associated graph the infinite structure of a structured system is characterized in terms of input output paths in the associated graph

typical convex program is very well posed in this paper we consider the collection of convex programming problems with inequality and equality constraints in which every problem of the collection is obtained by linear perturbations of the cost function and right hand side perturbation of the constraints while the core cost function and the left hand side constraint functions are kept fixed the main result shows that the set of the problems which are not well posed is porous in a certain strong sense our results concern both the infinite and finite dimensional case in the last case the conclusions are significantly sharper

the strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications for a locally optimal solution to the nonlinear semidefinite programming problem under robinson s constraint qualification the following conditions are proved to be equivalent the strong second order sufficient condition and constraint nondegeneracy the nonsingularity of clarke s jacobian of the karush kuhn tucker system the strong regularity of the karush kuhn tucker point and others

reduced state sequence estimation with set partitioning and decision feedback a reduced state sequence estimator for linear intersymbol interference channels is described the estimator uses a conventional viterbi algorithm with decision feedback to search a reduced state subset trellis that is constructed using set partitioning principles the complexity of maximum likelihood sequence estimation mlse due to the length of the channel memory and the size of the signal set is systematically reduced an error probability analysis shows that a good performance complexity tradeoff can be obtained in particular the results indicate that the required complexity to achieve the performance of mlse is independent of the size of the signal set for large enough signal sets simulation results are provided for two partial response systems a simple technique for quadrature partial response signaling qprs is described that eliminates the quasicatastrophic nature of the ml trellis

finite termination of the proximal point algorithm this paper concerns the notion of a sharp minimum on a set and its relationship to the proximal point algorithm we give several equivalent definitions of the property and use the notion to prove finite termination of the proximal point algorithm