intrinsic volumes of symmetric cones and applications in convex programming we express the probability distribution of the solution of a random standard gaussian instance of a convex cone program in terms of the intrinsic volumes and curvature measures of the reference cone we then compute the intrinsic volumes of the cone of positive semidefinite matrices over the real numbers over the complex numbers and over the quaternions in terms of integrals related to mehta s integral in particular we obtain a closed formula for the probability that the solution of a random standard gaussian semidefinite program has a certain rank

convergence of generalized aor iterative method for linear systems with strictly diagonally dominant matrices in this paper some improvements on darvishi and hessari on convergence of the generalized aor method for linear systems with diagonally dominant coefficient matrices appl math comput 176 2006 128 133 are presented for bounds of the spectral radius of l w r which is the iterative matrix of the generalized aor gaor method subsequently some new sufficient conditions for convergence of gaor method will be given which improve some results of darvishi and hessari on convergence of the generalized aor method for linear systems with diagonally dominant coefficient matrices appl math comput 176 2006 128 133

matrix sum of squares relaxations for robust semi definite programs we consider robust semi definite programs which depend polynomially or rationally on some uncertain parameter that is only known to be contained in a set with a polynomial matrix inequality description on the basis of matrix sum of squares decompositions we suggest a systematic procedure to construct a family of linear matrix inequality relaxations for computing upper bounds on the optimal value of the corresponding robust counterpart with a novel matrix version of putinar s sum of squares representation for positive polynomials on compact semi algebraic sets we prove asymptotic exactness of the relaxation family under a suitable constraint qualification if the uncertainty region is a compact polytope we provide a new duality proof for the validity of putinar s constraint qualification with an a priori degree bound on the polynomial certificates finally we point out the consequences of our results for constructing relaxations based on the so called full block s procedure which allows to apply recently developed tests in order to computationally verify the exactness of possibly small sized relaxations

on the convergence of the generalized aor method abstract we consider the generalized aor method presented by k r james we give some convergence theorems for various kinds of matrices namely when the coefficient matrix of the linear system is positive definite an h l or m matrix or strictly or irreducibly diagonally dominant

exact duals and short certificates of infeasibility and weak infeasibility in conic linear programming in conic linear programming in contrast to linear programming the lagrange dual is not an exact dual it may not attain its optimal value or there may be a positive duality gap the corresponding farkas lemma is also not exact it does not always prove infeasibility we describe exact duals and certificates of infeasibility and weak infeasibility for conic lps which are nearly as simple as the lagrange dual but do not rely on any constraint qualification some of our exact duals generalize the sdp duals of ramana and klep and schweighofer to the context of general conic lps some of our infeasibility certificates generalize the row echelon form of a linear system of equations they consist of a small trivially infeasible subsystem obtained by elementary row operations we prove analogous results for weakly infeasible systems we obtain some fundamental geometric corollaries an exact characterization of when the linear image of a closed convex cone is closed and an exact characterization of nice cones our infeasibility certificates provide algorithms to generate all infeasible conic lps over several important classes of cones and all weakly infeasible sdps in a natural class using these algorithms we generate a public domain library of infeasible and weakly infeasible sdps the status of our instances can be verified by inspection in exact arithmetic but they turn out to be challenging for commercial and research codes

characteristic invariant subspaces generated by a single vector abstract if f is an endomorphism of a finite dimensional vector space v over a field k then an invariant subspace x v is called hyperinvariant respectively characteristic if x is invariant under all endomorphisms respectively automorphisms that commute with f the characteristic hull of a subset w of v is defined to be the smallest characteristic subspace in v that contains w it is known that characteristic subspaces that are not hyperinvariant can only exist when k 2 in this paper we study subspaces x which are the characteristic hull of a single element in the case where k 2 we derive a necessary and sufficient condition such that x is hyperinvariant

a young eidson s type algorithm for complex p cyclic sor spectra abstract in a recent work of ours we have solved the problem of the minimization of the spectral radius of the iteration matrix of a p cyclic successive overrelaxation sor method for the solution of the linear system ax b when the matrix a is block p cyclic consistently ordered for what is known as the one point problem for any p 3 particular cases of the one point problem were solved by young varga kjellberg kredell russell and others in the present work we develop a theory using the results of our previous one and solve first the two point problem special cases of which were solved by wrigley eiermann niethammer ruttan noutsos and others secondly we generalize and extend our theory to cover the many point problem and develop a young eidson s type algorithm for its solution as possible application areas we mention among others the best block p cyclic repartitioning for the sor method are the solution of large scale systems arising in queueing network problems in markov analysis

semidefinite characterization of sum of squares cones in algebras we extend nesterov s semidefinite programming characterization of squared functional systems and faybusovich s abstraction to bilinear symmetric maps to cones of sum of squares elements in general abstract algebras using algebraic techniques such as isomorphism linear isomorphism tensor products sums and direct sums we show that many concrete cones are in fact sum of squares cones with respect to some algebra and thus are representable by the cone of positive semidefinite matrices we also consider nonnegativity with respect to a proper cone mathcal k and show that in some cases cones of mathcal k nonnegative functions are either sum of squares or at least semidefinite representable for example we show that some well known chebyshev systems when extended to euclidean jordan algebras induce cones that are semidefinite representable finally we will discuss some concrete examples and applications including minimum ellipsoid enclosing given space curves minimization of eigenvalues of po

on the implementation of interior point decomposition algorithms for two stage stochastic conic programs in this paper we develop a practical primal interior decomposition algorithm for two stage stochastic programming problems the framework of this algorithm is similar to the framework in s mehrotra and m g ozevin decomposition based interior point methods for two stage stochastic convex quadratic programs with recourse to appear in oper res s mehrotra and m g ozevin siam j optim 18 2007 pp 206 222 and g zhao math program 90 2001 pp 507 536 however their algorithm is altered in a simple yet fundamental way to achieve practical performance in particular this new algorithm weighs the log barrier terms in the second stage problems differently from the theoretical algorithms analyzed in s mehrotra and m g ozevin oper res to appear s mehrotra and m g ozevin siam j optim 18 2007 pp 206 222 and g zhao math program 90 2001 pp 507 536 we give a method for generating a suitable starting point a method for selecting a good starting barrier parameter a heuristic for first stage step length calculation without performing line searches and a method for adaptive addition of new scenarios over the course of the algorithm the decomposition algorithm is implemented to solve two stage stochastic conic programs with recourse whose underlying cones are cartesian products of linear second order and semidefinite cones the performance of primal decomposition method is studied on a set of randomly generated test problems as well as a two stage stochastic programming extension of the markowitz portfolio selection model the computational results show that an efficient and stable implementation of the primal decomposition method is possible these results also show that in problems with a large number of scenarios the adaptive addition of scenarios can yield computational savings of up to 80

characteristic subspaces and hyperinvariant frames abstract let f be an endomorphism of a finite dimensional vector space v over a field k an f invariant subspace is called hyperinvariant respectively characteristic if it is invariant under all endomorphisms respectively automorphisms that commute with f we assume k 2 since all characteristic subspaces are hyperinvariant if k 2 the hyperinvariant hull w h of a subspace w of v is defined to be the smallest hyperinvariant subspace of v that contains w the hyperinvariant kernel w h of w is the largest hyperinvariant subspace of v that is contained in w and the pair w h w h is the hyperinvariant frame of w in this paper we study hyperinvariant frames of characteristic non hyperinvariant subspaces w we show that all invariant subspaces in the interval w h w h are characteristic we use this result for the construction of characteristic non hyperinvariant subspaces

optimal p cyclic sor for complex spectra abstract we consider the successive overrelaxation sor method for the solution of a linear system ax b when the matrix a has a block p p partitioned p cyclic form and its associated block jacobi matrix jp is weakly cyclic of index p following the pioneering work by young and varga in the 1950s many researchers have considered various cases for the spectrum jp and have determined optimal values for the relaxation factor 0 2 so that the sor method converges as fast as possible after the most recent work on the best block p cyclic repartitioning and that on the solution of large scale systems arising in queueing network problems in markov analysis the optimization of the convergence of the p cyclic sor for more complex spectra jp has become more demanding here we state the one point problem for the general p cyclic complex sor case the existence and the uniqueness of its solution are established by analyzing and developing further the theory of the associated hypocycloidal curves for the determination of the optimal parameter s an algorithm is presented and a number of illustrative numerical examples are given

on the equivalence of extrapolation and richardson s iteration and its applications abstract sometime ago two articles g opfer g schober richardson s iteration for nonsymmetric matrices linear algebra appl 58 1984 343 361 a hadjidimos the optimal solution to the problem of complex extrapolation of a first order scheme linear algebra appl 62 1984 241 261 appeared the same year in this journal solving the problem of optimizing the convergence factor of the stationary richardson s iteration i e x m 1 x m a ax m b m 0 1 2 for the solution of the system ax b a c n n det a 0 b c n in opfer and schober g opfer g schober richardson s iteration for nonsymmetric matrices linear algebra appl 58 1984 343 361 when the spectrum of a was contained in a certain compact set s 1 not including the origin the optimal parameter a was determined in hadjidimos a hadjidimos the optimal solution to the problem of complex extrapolation of a first order scheme linear algebra all 62 1984 241 261 for the solution of ax b the consistent linear scheme x m 1 tx m c m 0 1 2 for which the spectrum of t was contained in a certain compact convex set s 2 not including the point 1 was considered and the optimal extrapolation parameter of its extrapolation scheme i e x m 1 1 i t x m c m 0 1 2 was determined because the approach in obtaining the optimal parameters in the two articles was quite different it seems it was difficult for any one to notice their equivalence and therefore to use them even in trivial cases in this article we show the equivalence of the optimal results in opfer and schober g opfer g schober richardson s iteration for nonsymmetric matrices linear algebra appl 58 1984 343 361 and hadjidimos a hadjidimos the optimal solution to the problem of complex extrapolation of a first order scheme linear algebra all 62 1984 241 261 explore them further obtain new results and give applications for cases which may occur in practice finally based on observations regarding the optimal two cyclic successive overrelaxation sor and accelerated overrelaxation aor methods in two real case problems we solve the problem of optimal two cyclic aor in the more general complex case

projective re normalization for improving the behavior of a homogeneous conic linear system in this paper we study the homogeneous conic system f ax 0 x in c setminus 0 we choose a point bar s in rm int c that serves as a normalizer and consider computational properties of the normalized system f bar s ax 0 bar s t x 1 x in c we show that the computational complexity of solving f via an interior point method depends only on the complexity value vartheta of the barrier for c and on the symmetry of the origin in the image set h bar s ax bar s tx 1 x in c where the symmetry of 0 in h bar s is rm sym 0 h bar s rm max alpha y in h bar s rightarrow alpha y in h bar s interior point iterations in order to improve the theoretical and practical computation of a solution of f we next present a general theory for projective re normalization of the feasible region f bar s and the image set h bar s and prove the existence of a normalizer bar s such that rm sym 0 h bar s ge 1 m provided that f has an interior solution we develop a methodology for constructing a normalizer bar s such that rm sym 0 h bar s ge 1 m with high probability based on sampling on a geometric random walk with associated probabilistic complexity analysis while such a normalizer is not itself computable in strongly polynomial time the normalizer will yield a conic system that is solvable in o sqrt vartheta rm ln m vartheta iterations which is strongly polynomial time finally we implement this methodology on randomly generated homogeneous linear programming feasibility problems constructed to be poorly behaved our computational results indicate that the projective re normalization methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems for instance we observe a 46 decrease in average ipm iterations for 100 randomly generated poorly behaved problem instances of dimension 1 000 5 000

exposed faces of semidefinitely representable sets a linear matrix inequality lmi is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite motivated by the fact that diagonal lmis define polyhedra the solution set of an lmi is called a spectrahedron linear images of spectrahedra are called semidefinitely representable sets part of the interest in spectrahedra and semidefinitely representable sets arises from the fact that one can efficiently optimize linear functions on them by semidefinite programming such as one can do on polyhedra by linear programming it is known that every face of a spectrahedron is exposed this is also true in the general context of rigidly convex sets we study the same question for semidefinitely representable sets lasserre proposed a moment matrix method to construct semidefinite representations for certain sets our main result is that this method can work only if all faces of the considered set are exposed this necessary condition complements sufficient conditions recently proved by lasserre helton and nie

universal duality in conic convex optimization given a primal dual pair of linear programs it is well known that if their optimal values are viewed as lying on the extended real line then the duality gap is zero unless both problems are infeasible in which case the optimal values are and in contrast for optimization problems over nonpolyhedral convex cones a nonzero duality gap can exist when either the primal or the dual is feasible for a pair of dual conic convex programs we provide simple conditions on the constraint matrices and cone under which the duality gap is zero for every choice of linear objective function and constraint right hand side we refer to this property as universal duality our conditions possess the following properties i they are necessary and sufficient in the sense that if and only if they do not hold the duality gap is nonzero for some linear objective function and constraint right hand side ii they are metrically and topologically generic and iii they can be verified by solving a single conic convex program we relate to universal duality the fact that the feasible sets of a primal convex program and its dual cannot both be bounded unless they are both empty finally we illustrate our theory on a class of semidefinite programs that appear in control theory applications

discretization method for semi definite programming semi definite programs are convex optimization problems arising in a wide variety of applications and the extension of linear programming most methods for linear programming have been generalized to semi definite programs this paper discusses the discretization method in semi definite programming the convergence and the convergent rate of error between the optimal value of the semi definite programming problems and the optimal value of the discretized problems are obtained an approximately optimal division is given under certain assumptions with the significance of the convergence property the duality result in semi definite programs is proved in a simple way which is different from the other common proofs

on the foundations of parameter estimation for generalized partial linear models with b splines and continuous optimization generalized linear models are widely used in statistical techniques as an extension generalized partial linear models utilize semiparametric methods and augment the usual parametric terms with a single nonparametric component of a continuous covariate in this paper after a short introduction we present our model in the generalized additive context with a focus on the penalized maximum likelihood and the penalized iteratively reweighted least squares p irls problem based on b splines which is attractive for nonparametric components then we approach solving the p irls problem using continuous optimization techniques they have come to constitute an important complementary approach alternative to the penalty methods with flexibility for choosing the penalty parameter adaptively in particular we model and treat the constrained p irls problem by using the elegant framework of conic quadratic programming the method is illustrated using a small numerical example

orbits of invariant subspaces of algebraic linear operators abstract let e be a vector space over a field k and a an algebraic k linear operator on e we describe the orbits under the action of the commutant of a and of the group of the invertible elements of the commutant of a on the lattice of a invariant subspaces

extremum seeking control of state constrained nonlinear systems an extremum seeking control problem is posed for a class of nonlinear systems with unknown dynamical parameters whose states are subject to convex pointwise inequality constraints using a barrier function approach an adaptive method is proposed for generating setpoints online which converge to the feasible minimizer of a convex objective function containing the unknown dynamic parameters a tracking controller regulates system states to the generated setpoint via state feedback while maintaining feasibility of the state constraints a simulation example demonstrates application of the method

generic spectrahedral shadows spectrahedral shadows are projections of linear sections of the cone of positive semidefinite matrices we characterize the polynomials that vanish on the boundaries of these convex sets when both the section and the projection are generic

generalized aor methods for linear complementarity problem abstract in this paper we firstly establish a class of generalized aor gaor methods for solving a linear complementarity problem lcp m q whose special case reduces to generalized sor gsor method then some sufficient conditions for convergence of the gaor and gsor methods are presented when the system matrix m is an h matrix m matrix and a strictly or irreducible diagonally dominant matrix moreover when m is an l matrix we discuss the monotone convergence of the new methods lastly we report some computational results with the proposed methods

on the connection of facially exposed and nice cones abstract a closed convex cone k in a finite dimensional euclidean space is called nice if the set k f is closed for all f faces of k where k is the dual cone of k and f is the orthogonal complement of the linear span of f the niceness property plays a role in the facial reduction algorithm of borwein and wolkowicz and the question of whether the linear image of the dual of a nice cone is closed also has a simple answer we prove several characterizations of nice cones and show a strong connection with facial exposedness we prove that a nice cone must be facially exposed conversely facial exposedness with an added condition implies niceness we conjecture that nice and facially exposed cones are actually the same and give supporting evidence

sous espaces invariants d un operateur nilpotent sur un espace de banach resume soit a un operateur lineaire continu nilpotent sur un espace de banach reel ou complexe nous montrons que pour tout sous espace invariant separable m de a il existe un commutant de a dont l image est dense dans m

an interior point trust region algorithm for general symmetric cone programming an interior point trust region algorithm is proposed for minimizing a general non convex quadratic objective function in the intersection of a symmetric cone and an affine subspace the algorithm uses a trust region model to ensure descent on a suitable merit function global first order and second order convergence results are proved numerical results are presented

an exact duality theory for semidefinite programming and its complexity implications in this paper we present a new and more complete duality for semidefinite programming sdp with the following features begin itemize item this dual is an explicit semidefinite program whose number of variables and the coefficient bitlengths are polynomial in those of the primal item if the primal is feasible then it is bounded if and only if the dual is feasible item the duality gap ie the difference between the primal and the dual objective function values is zero whenever the primal is feasible and bounded also in this case the dual attains its optimum item it yields a precise farkas lemma for semidefinite feasibility systems ie characterization of the it infeasibility of a semidefinite inequality in terms of the it feasibility of another polynomial size semidefinite inequality end itemize note that the standard duality for linear programming satisfies all of the above features but no such duality theory was previously known for sdp without slater like conditions being assumed then we apply the dual to derive certain complexity results for semidefinite programming problems the decision problem of semidefinite feasibility sdfp ie that of determining if a given semidefinite inequality system is feasible is the central problem of interest the complexity of sdfp is unknown but we show the following 1 in the turing machine model sdfp is not np complete unless np co np 2 in the real number model of blum shub and smale cite bss sdfp is in np cap co np we then give polynomial reductions from the following problems to sdfp 1 checking whether an sdp is bounded 2 checking whether a feasible and bounded sdp attains the optimum 3 checking the optimality of a feasible solution

tractable approximations to robust conic optimization problems in earlier proposals the robust counterpart of conic optimization problems exhibits a lateral increase in complexity i e robust linear programming problems lps become second order cone problems socps robust socps become semidefinite programming problems sdps and robust sdps become np hard we propose a relaxed robust counterpart for general conic optimization problems that a preserves the computational tractability of the nominal problem specifically the robust conic optimization problem retains its original structure i e robust lps remain lps robust socps remain socps and robust sdps remain sdps and b allows us to provide a guarantee on the probability that the robust solution is feasible when the uncertain coefficients obey independent and identically distributed normal distributions

is a cn n a general h matrix abstract h matrices play an important role in the theory and applications of numerical linear algebra so it is very useful to know whether a given matrix a c n n usually the coefficient of a complex linear system of algebraic equations or of a linear complementarity problem a r n n with a ii 0 for i 1 2 n in this case is an h matrix then most of the classical iterative methods for the solution of the problem at hand converge in recent years the set of h matrices has been extended to what is now known as the set of general h matrices and a partition of this set in three different classes has been made the main objective of this work is to develop an algorithm that will determine the h matrix character and will identify the class to which a given matrix a c n n belongs in addition some results on the classes of general h matrices and a partition of the non h matrix set are presented

on the second order feasibility cone primal dual representation and efficient projection we study the second order feasibility cone mathcal f y in mathbb r n my leq g ty for given data m g we construct a new representation for this cone and its dual based on the spectral decomposition of the matrix m tm gg t this representation is used to efficiently solve the problem of projecting an arbitrary point x in mathbb r n onto mathcal f min y y x my leq g ty which aside from theoretical interest also arises as a necessary subroutine in the rescaled perceptron algorithm we develop a method for solving the projection problem to an accuracy varepsilon whose computational complexity is bounded by o mn 2 n ln ln 1 varepsilon n ln ln 1 min operatorname width mathcal f operatorname width mathcal f operations here operatorname width mathcal f and operatorname width mathcal f denote the width of mathcal f and mathcal f respectively we also perform computational tests that indicate that the method is extremely efficient in practice

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

the convex positivstellensatz in a free algebra abstract given a monic linear pencil l in g variables let p l p l n n n where p l n x s n g l x 0 and s n g is the set of g tuples of symmetric n n matrices because l is a monic linear pencil each p l n is convex with interior and conversely it is known that convex bounded noncommutative semialgebraic sets with interior are all of the form p l the main result of this paper establishes a perfect noncommutative nichtnegativstellensatz on a convex semialgebraic set namely a noncommutative matrix valued polynomial p is positive semidefinite on p l if and only if it has a weighted sum of squares representation with optimal degree bounds p s s j finite f j l f j where s f j are matrices of noncommutative polynomials of degree no greater than deg p 2 this noncommutative result contrasts sharply with the commutative setting where there is no control on the degrees of s f j and assuming only p nonnegative as opposed to p strictly positive yields a clean positivstellensatz so seldom that such cases are noteworthy

strong duality for semidefinite programming it is well known that the duality theory for linear programming lp is powerful and elegant and lies behind algorithms such as simplex and interior point methods however the standard lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps semidefinite linear programming sdp is a generalization of lp where the nonnegativity constraints are replaced by a semidefiniteness constraint on the matrix variables there are many applications e g in systems and control theory and combinatorial optimization however the lagrangian dual for sdp can have a duality gap we discuss the relationships among various duals and give a unified treatment for strong duality in semidefinite programming these duals guarantee strong duality i e a zero duality gap and dual attainment this paper is motivated by the recent paper by ramana where one of these duals is introduced

sur l orbite d un espace de banach sous l action du commutant d un operateur nilpotent resume soit a un operateur nilpotent sur un espace de banach e nous demontrons que l orbite de e par l action du commutant de a contient les f lat a qui sont de dimension finie

decomposition based interior point methods for two stage stochastic convex quadratic programs with recourse zhao showed that the log barrier associated with the recourse function of two stage stochastic linear programs behaves as a strongly self concordant barrier and forms a self concordant family on the first stage solutions in this paper we show that the recourse function is also strongly self concordant and forms a self concordant family for the two stage stochastic convex quadratic programs with recourse this allows us to develop bender s decomposition based linearly convergent interior point algorithms an analysis of such an algorithm is given in this paper

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

semigroups of relations and subdifferentials of the minimal time function and of the geodesic distance given a closed subset s of a banach space x we study the minimal time function to reach a point x of x starting from s we relate this problem to the study of the geodesic distance from x to s associated with a riemannian metric or a finsler metric it appears that both cases can be treated simultaneously and yield solutions to a hamilton jacobi equation we detect simple links between the general framework we propose and multivalued semigroups new concepts of infinitesimal generators of such semigroups are considered

the matrix analogue of the scalar aor iterative method the accelerated overrelaxation aor and the generalized aor gaor iterative methods for the solution of linear systems of algebraic equations a x b a c n i n det a 0 b c n have been around for about four decades and a plethora of variations of them have been proposed in this work a novel algorithm is introduced the matrix analogue of the aor maaor iterative method which is analysed and studied the maaor method generalizes both the aor and the gaor sufficient convergence conditions for the gaor method are determined when the coefficient matrix a of the linear system to be solved is a hermitian matrix with positive diagonal elements similarly sufficient convergence conditions for the maaor method are determined when a is a nonsingular h matrix the new convergence conditions are the most general ones so far numerical examples are presented in support of the theory developed

an exact duality theory for semidefinite programming based on sums of squares farkas lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities in semidefinite programming such linear certificates only exist for strongly infeasible linear matrix inequalities we provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry a linear matrix inequality ax succeq 0 is infeasible if and only if 1 lies in the quadratic module associated to a we also present a new exact duality theory for semidefinite programming motivated by the real radical and sums of squares certificates from real algebraic geometry

on the closedness of the linear image of a closed convex cone when is the linear image of a closed convex cone closed we present very simple and intuitive necessary conditions that 1 unify and generalize seemingly disparate classical sufficientconditions such as polyhedrality of the cone and slater type conditions 2 are necessary and sufficient when the dual cone belongs to a class that we call nice cones nice cones subsume all cones amenable to treatment by efficient optimization algorithms for instance polyhedral semidefinite and p cones and 3 provide similarly attractive conditions for an equivalent problem the closedness of the sum of two closed convex cones

augmented self concordant barriers and nonlinear optimization problems with finite complexity in this paper we study special barrier functions for convex cones which are the sum of a self concordant barrier for the cone and a positive semidefinite quadratic form we show that the central path of these augmented barrier functions can be traced with linear speed we also study the complexity of finding the analytic center of the augmented barrier a problem that has some interesting applications we show that for some special classes of quadratic forms and some convex cones the computation of the analytic center requires an amount of operations independent of the particular data set we argue that these problems form a class that is endowed with a property which we call finite polynomial complexity

a homogeneous interior point algorithm for nonsymmetric convex conic optimization a homogeneous interior point algorithm for solving nonsymmetric convex conic optimization problems is presented starting each iteration from the vicinity of the central path the method steps in the approximate tangent direction and then applies a correction phase to locate the next well centered primal dual point features of the algorithm include that it makes use only of the primal barrier function that it is able to detect infeasibilities in the problem and that no phase i method is needed we prove convergence to epsilon accuracy in mathcal o sqrt nu log 1 epsilon o log 1 iterations to improve performance the algorithm employs a new runge kutta type second order search direction suitable for the general nonsymmetric conic problem moreover quasi newton updating is used to reduce the number of factorizations needed implemented so that data sparsity can still be exploited extensive and promising computational results are presented for the p p cone problem the facility location problem entropy maximization problems and geometric programs all formulated as nonsymmetric convex conic optimization problems

barrier function based model predictive control a new formulation of nonlinear model predictive control mpc is developed by including a weighted barrier function in the control objective while the barrier ensures that inequality constraints are strictly satisfied it also provides a smooth transition between points in the interior and those on the boundary of the constraint set in addition the resulting optimisation problem to be solved at each control step is effectively unconstrained and thus amenable to elegant optimisation techniques the barrier must satisfy certain conditions in order that the state converges to the origin and we show how to construct such a barrier conventional mpc may be seen as a limiting case of the new class as the barrier weighting itself approaches zero we pay particular attention to the novel approach of fixing the weighting parameter to some positive value possibly large and observe that this provides a degree of controller caution near constraint boundaries we construct an ellipsoidal invariant set by exploiting the geometry of self concordant functions and show nominal closed loop stability for this class of controllers under full state feedback

exact duality in semidefinite programming based on elementary reformulations in semidefinite programming sdp unlike in linear programming farkas lemma may fail to prove infeasibility here we obtain an exact short certificate of infeasibility in sdp by an elementary approach we reformulate any equality constrained semidefinite system using only elementary row operations and rotations when a system is infeasible the reformulated system is trivially infeasible when a system is feasible the reformulated system has strong duality with its lagrange dual for all objective functions as a corollary we obtain algorithms to generate the constraints of all infeasible sdps and the constraints of all feasible sdps with a fixed rank maximal solution our elementary reformulations can be constructed either by a direct method or by adapting the waki muramatsu facial reduction algorithm in different language the reformulations provide a standard form of spectrahedra to easily verify either their emptiness or a tight upper bound on the rank of feasible solutions

regularizing the abstract convex program abstract characterizations of optimality for the abstract convex program inf p x g x s x p where s is an arbitrary convex cone in a finite dimensional space is a convex set and p and g are respectively convex and s convex on were given in 10 these characterizations hold without any constraint qualification they use the minimal cone s f of p and the cone of directions of constancy d g s f in the faithfully convex case these cones can be used to regularize p i e transform p into an equivalent program p r for which slater s condition holds we present an algorithm that finds both s f and d g s f the main step of the algorithm consists in solving a particular complementarity problem we also present a characterization of optimality for p in terms of the cone of directions of constancy of a convex functional d g rather than d g s f

invariance and efficiency of convex representations we consider two notions for the representations of convex cones g representation and lifted g representation the former represents a convex cone as a slice of another the latter allows in addition the usage of auxiliary variables in the representation we first study the basic properties of these representations we show that some basic properties of convex cones are invariant under one notion of representation but not the other in particular we prove that lifted g representation is closed under duality when the representing cone is self dual we also prove that strict complementarity of a convex optimization problem in conic form is preserved under g representations then we move to study efficiency measures for representations we evaluate the representations of homogeneous convex cones based on the smoothness of the transformations mapping the central path of the representation to the central path of the represented optimization problem