transient behavior analysis of vibrationally controlled nonlinear parabolic systems with neumann boundary conditions the transient behavior of vibrationally controlled distributed parameter systems governed by parabolic partial differential equations pdes with neumann boundary conditions is analyzed the analysis uses a certain mapping and a time invariant pde whose trajectories under this mapping yield the approximate moving averages along the trajectories of the vibrationally controlled system this approach considerably enhances the understanding of the behavior of vibrationally controlled distributed parameter systems and facilitates a selection of the parameters of stabilizing vibrations that ensure transient behavior with more desirable properties

superlinear convergence of an infeasible predictor corrector path following interior point algorithm for a semidefinite linear complementarity problem using the helmberg kojima monteiro direction an interior point method ipm defines a search direction at each interior point of a region these search directions form a direction field which in turn gives rise to a system of ordinary differential equations odes the solutions of the system of odes can be viewed as underlying paths in the interior of the region in c k sim and g zhao math program ser a 110 2007 pp 475 499 these off central paths are shown to be well defined analytic curves and any of their accumulation points is a solution to a given monotone semidefinite linear complementarity problem sdlcp the study of these paths provides a way to understand how iterates generated by an interior point algorithm behave in this paper we give a sufficient condition using these off central paths that guarantees superlinear convergence of a predictor corrector path following interior point algorithm for sdlcp using the helmberg kojima monteiro hkm direction this sufficient condition is implied by a currently known sufficient condition for superlinear convergence using this sufficient condition we show that for any linear semidefinite feasibility problem superlinear convergence using the interior point algorithm with the hkm direction can be achieved for a suitable starting point we work under the assumption of strict complementarity

vibrational control of nonlinear systems vibrational stabilizability in this work the theory of vibrational control for nonlinear systems is developed the present paper gives criteria for stabilizability of nonlinear systems by linear multiplicative vibrations by almost periodic forcing and by vector additive vibrations illustrative examples are also considered in a subsequent paper the problem of vibrational controllability and transient behavior will be addressed

shallow deep and very deep cuts in the analytic center cutting plane method in this paper we show that the cut does not need to go through the query point it can be deep or shallow the primal framework leads to a simple analysis of the potential variation which shows that the inequality needed for convergence of the algorithm is in fact attained at the first iterate of the feasibility step

a potential reduction algorithm allowing column generation using the dantzig wolfe decomposition technique a potential reduction algorithm allowing column generation for the linear feasibility lf problem is developed the point of departure is a simple containing polytope and its analytic center in each iteration an inequality violated at the current center is selected used to cut the polytope and then used to find the new center for the shrunken polytope the potential value associated with the containing polytope is reduced by a constant at each step and the algorithm is terminated in a polynomial time that depends only on the number and data length of the inequalities generated in the iterative process

a probabilistic analytic center cutting plane method for feasibility of uncertain lmis many robust control problems can be formulated in abstract form as convex feasibility programs where one seeks a solution x that satisfies a set of inequalities of the form f f x d 0 d d this set typically contains an infinite and uncountable number of inequalities and it has been proved that the related robust feasibility problem is numerically hard to solve in general in this paper we discuss a family of cutting plane methods that solve efficiently a probabilistically relaxed version of the problem specifically under suitable hypotheses we show that an analytic center cutting plane scheme based on a probabilistic oracle returns in a finite and pre specified number of iterations a solution x which is feasible for most of the members of f except possibly for a subset having arbitrarily small probability measure

a log barrier method with benders decomposition for solving two stage stochastic linear programs an algorithm incorporating the logarithmic barrier into the benders decomposition technique is proposed for solving two stage stochastic programs basic properties concerning the existence and uniqueness of the solution and the underlying path are studied when applied to problems with a finite number of scenarios the algorithm is shown to converge globally and to run in polynomial time

chaotic behavior of the affine scaling algorithm for linear programming the affine scaling algorithm for linear programming involves a step size parameter t that must be chosen in the interval 0 1 it is known that the algorithm converges to an optimal solution for values of t le 2 3 in this paper we examine the behavior of the algorithm for values of t 2 3 we show that for certain values of t in this range the algorithm can exhibit chaotic behavior

an interior point cutting plane method for the convex feasibility problem with second order cone inequalities the convex feasibility problem in general is a problem of finding a point in a convex set that contains a full dimensional ball and is contained in a compact convex set we assume that the outer set is described by second order cone inequalities and propose an analytic center cutting plane technique to solve this problem we discuss primal and dual settings simultaneously two complexity results are reported the complexity of restoration procedure and complexity of the overall algorithm we prove that an approximate analytic center is updated after adding a second order cone cut socc ino 1 newton step and that the analytic center cutting plane method accpm with socc is a fully polynomial algorithm

complexity analysis of an interior cutting plane method for convex feasibility problems we further analyze the convergence and the complexity of a dual column generation algorithm for solving general convex feasibility problems defined by a separation oracle the oracle is called at an approximate analytic center of the set given by the intersection of the linear inequalities which are the previous answers of the oracle we show that the algorithm converges in finite time and is in fact a fully polynomial approximation algorithm provided that the feasible region has a nonempty interior

a cutting plane algorithm for convex programming that uses analytic centers anoracle for a convex sets n accepts as input any pointz in n and ifz s then it returns yes while ifz s then it returns no along with a separating hyperplane we give a new algorithm that finds a feasible point ins in cases where an oracle is available our algorithm uses the analytic center of a polytope as test point and successively modifies the polytope with the separating hyperplanes returned by the oracle the key to establishing convergence is that hyperplanes judged to be unimportant are pruned from the polytope if a ball of radius 2 l is contained ins ands is contained in a cube of side 2l 1 then we can show our algorithm converges after o nl2 iterations and performs a total of o n4l3 tnl2 arithmetic operations wheret is the number of arithmetic operations required for a call to the oracle the bound is independent of the number of hyperplanes generated in the algorithm an important application in which an oracle is available is minimizing a convex function overs

bounded error parameter estimation a sequential analytic center approach a sequential analytic center approach for bounded error parameter estimation is proposed the authors show that the analytic center minimizes the logarithmic average output error among all the estimates within the membership set and is a maximum likelihood estimator for a class of noise density functions which include parabolic densities and approximations of truncated gaussian they also show that the analytic center is easily computable for both offline and online problems with a sequential algorithm the convergence proof of this sequential algorithm is obtained and moreover it is shown that the complexity in terms of the maximum number of newton iterations is linear in the number of observed data points

a multiple cut analytic center cutting plane method for semidefinite feasibility problems we consider the problem of finding a point in a nonempty bounded convex body gamma in the cone of symmetric positive semidefinite matrices cal s m assume that gamma is defined by a separating oracle which for any given m ti m symmetric matrix hat y either confirms that hat y in gamma or returns several selected cuts i e a number of symmetric matrices ai i 1 p p le p max such that gamma is in the polyhedron y in cal s m mid a i bullet y le a i bullet hat y i 1 ldots p we present a multiple cut analytic center cutting plane algorithm starting from a trivial initial point the algorithm generates a sequence of positive definite matrices which are approximate analytic centers of a shrinking polytope in cal s m the algorithm terminates with a point in gamma within o m 3p max epsilon 2 newton steps to leading order where epsilon is the maximum radius of a ball contained in gamma

sequential randomized algorithms for robust convex optimization sequential randomized algorithms are considered for robust convex optimization which minimizes a linear objective function subject to a parameter dependent convex constraint employing convex optimization and random sampling of parameter these algorithms enable us to obtain a suboptimal solution within reasonable computational time the suboptimal solution is feasible in a probabilistic sense and the suboptimal value belongs to an interval which contains the optimal value the maximum of the interval is the optimal value of the robust convex optimization plus a specified tolerance on the other hand its minimum is the optimal value of the chance constrained optimization which is a probabilistic relaxation of the robust convex optimization with high probability

the volumetric barrier for semidefinite programming we consider the volumetric barrier for semidefinite programming or generalized volumetric barrier as introduced by nesterov and nemirovskii we extend several fundamental properties of the volumetric barrier for a polyhedral set to the semidefinite case our analysis facilitates a simplified proof of self concordance for the semidefinite volumetric barrier as well as for the combined volumetric logarithmic barrier for semidefinite programming for both of these barriers we obtain self concordance parameters equal to those previously shown to hold in the polyhedral case

volumetric path following algorithms for linear programming we consider the construction of small step path following algorithms using volumetric and mixed volumetric logarithmic barriers we establish quadratic convergence of a volumetric centering measure using pure newton steps enabling us to use relatively standard proof techniques for several subsequently needed results using a mixed volumetric logarithmic barrier we obtain an o n 1 4 m 1 4 l iteration algorithm for linear programs withn variables andm inequality constraints providing an alternative derivation for results first obtained by vaidya and atkinson in addition we show that the same iteration complexity can be attained while holding the work per iteration to o n 2 m as opposed to o nm 2 operations by avoiding use of the true hessian of the volumetric barrier our analysis also provides a simplified proof of self concordancy of the volumetric and mixed volumetric logarithmic barriers originally due to nesterov and nemirovskii

a nonlinear analytic center cutting plane method for a class of convex programming problems a cutting plane algorithm for minimizing a convex function subject to constraints defined by a separation oracle is presented the algorithm is based on approximate analytic centers the nonlinearity of the objective function is taken into account yet the feasible region is approximated by a containing polytope this containing polytope is regularly updated by adding a new cut through a test point each test point is an approximate analytic center of the intersection of a containing polytope and a level set of the nonlinear objective function we establish the complexity of the algorithm our complexity estimate is given in terms of the problem dimension the desired accuracy of an approximate solution and other parameters that depend on the geometry of a specific instance of the problem

vibrational feedback control of time delay systems this paper applies vibrational feedback control to time lag systems averaging theory for differential delay systems is presented and then applied to aid in the design of periodic controllers both stabilization and transient issues are discussed an illustrative example is given which demonstrates that the proposed controller 1 provides superior robustness properties in comparison to time invariant controllers when applied correctly 2 is not robust with respect to unmodeled delays and 3 does not have zero placement capabilities in the sense defined in the paper

complexity estimates of some cutting plane methods based on the analytic barrier in this paper we establish the efficiency estimates for two cutting plane methods based on the analytic barrier we prove that the rate of convergence of the second method is optimal uniformly in the number of variables we present a modification of the second method in this modified version each test point satisfies an approximate centering condition we also use the standard strategy for updating approximate hessians of the logarithmic barrier function we prove that the rate of convergence of the modified scheme remains optimal and demonstrate that the number of newton steps in the auxiliary minimization processes is bounded by an absolute constant we also show that the approximate hessian strategy significantly improves the total arithmetical complexity of the method

a regularity theory for variational problems with higher order derivatives we consider problems in the calculus of variations in one independent variable and where the lagrangian involves derivatives up to order n n 1 existence theory supplies mild hypotheses under which there are minimizers for such problems but they need to be strengthened for standard necessary conditions to apply for problems with n 1 this paper initiates investigation of regularity properties and associated necessary conditions which obtain strictly under the hypotheses of existence theory it is shown that the nth derivative of a minimizer is locally essentially bounded off a closed set of zero measure the set of points of bad behaviour additional hypotheses are shown to exclude occurrence of points of bad behaviour finally a counter example suggests respects in which problems with n 1 exhibit pathologies not present in the n i case

solving combinatorial optimization problems using karmakar s algorithm we describe a cutting plane algorithm for solving combinatorial optimization problems the primal projective standard form variant of karmarkar s algorithm for linear programming is applied to the duals of a sequence of linear programming relaxations of the combinatorial optimization problem

ellipsoidal approximations of convex sets based on the volumetric barrier let c rn be a convex set we assume that x 1 for all x c and that c contains a ball of radius 1 r for x rn r r and b and n n symmetric positive definite matrix let ex b r y y xtby x r2 a rounding of c is an ellipsoid ex b r such that ex b r c ex b r in the case that c is characterized by a separation oracle it is well known that an on3 2 rounding of c can be obtained using the shallow cut ellipsoid method in on3 lnnr oracle calls we show that a modification of the volumetric cutting plane method obtains an on3 2 rounding of c in on2 lnnr oracle calls we also consider the problem of obtaining an on rounding of c when c has an explicit polyhedral description our analysis uses a new characterization of circumscribing ellipsoids centered at or near the volumetric center of a polyhedral set

a cutting plane method from analytic centers for stochastic programming the stochastic linear programming problem with recourse has a dual block angular structure it can thus be handled by benders decomposition or by kelley s method of cutting planes equivalently the dual problem has a primal block angular structure and can be handled by dantzig wolfe decomposition the two approaches are in fact identical by duality here we shall investigate the use of the method of cutting planes from analytic centers applied to similar formulations the only significant difference form the aforementioned methods is that new cutting planes or columns by duality will be generated not from the optimum of the linear programming relaxation but from the analytic center of the set of localization

on vaidya s volumetric cutting plane method for convex programming we describe a simplified and strengthened version of vaidya s volumetric cutting plane method for finding a point in a convex set x1d49e rn at each step the algorithm has a system of linear inequality constraints which defines a polyhedron x1d4ab x1d49e and an interior point x x1d4ab the algorithm then either drops one constraint or calls an oracle to check if x x1d49e and if not obtain a new constraint that separates x from x1d49e following the addition or deletion of a constraint the algorithm takes a small number of newton steps for the volumetric barrier v progress of the algorithm is measured in terms of changes in v the algorithm is terminated when either it is discovered that x x1d49e or v becomes large enough to demonstrate that the volume of x1d49e must be below some prescribed amount the complexity of the algorithm compares favorably with that of the ellipsoid method especially in terms of the number of calls to the separation oracle compared to vaidya s original analysis we decrease the total number

complexity analysis of the analytic center cutting plane method that uses multiple cuts we analyze the complexity of the analytic center cutting plane or column generation algorithm for solving general convex problems defined by a separation oracle the oracle is called at the analytic center of a polytope which contains a solution set and is given by the intersection of the linear inequalities previously generated from the oracle if the center is not in the solution set separating hyperplanes will be placed through the center to shrink the containing polytope while the complexity result has been recently established for the algorithm when one cutting plane is placed in each iteration the result remains open when multiple cuts are added moreover adding multiple cuts actually is a key to practical effectiveness in solving many problems and it presents theoretical difficulties in analyzing cutting plane methods in this paper we show that the analytic center cutting plane algorithm with multiple cuts added in each iteration still is a fully polynomial approximation algorithm

towards a practical volumetric cutting plane method for convex programming we consider the volumetric cutting plane method for finding a point in a convex set cal c subset re n that is characterized by a separation oracle we prove polynomiality of the algorithm with each added cut placed directly through the current point and show that this central cut version of the method can be implemented using no more than 25n constraints at any time

a new algorithm for minimizing convex functions over convex sets let s subseteq mathbb r n be a convex set for which there is an oracle with the following property given any pointz n the oracle returns a yes ifz s whereas ifz s then the oracle returns a no together with a hyperplane that separatesz froms the feasibility problem is the problem of finding a point ins the convex optimization problem is the problem of minimizing a convex function overs we present a new algorithm for the feasibility problem the notion of a volumetric center of a polytope and a related ellipsoid of maximum volume inscribable in the polytope are central to the algorithm our algorithm has a significantly better global convergence rate and time complexity than the ellipsoid algorithm the algorithm for the feasibility problem easily adapts to the convex optimization problem

decomposition and nondifferentiable optimization with the projective algorithm this paper deals with an application of a variant of karmarkar s projective algorithm for linear programming to the solution of a generic nondifferentiable minimization problem this problem is closely related to the dantzig wolfe decomposition technique used in large scale convex programming the proposed method is based on a column generation technique defining a sequence of primal linear programming maximization problems associated with each problem one defines a weighted potential function which is minimized using a variant of the projective algorithm when a point close to the minimum of the potential function is reached a corresponding point in the dual space is constructed which is close to the analytic center of a polytope containing the solution set of the nondifferentiable optimization problem an admissible cut of the polytope corresponding to a new supporting hyperplane of the epigraph of the function to minimize is then generated at this approximate analytic center in the primal space this new cut translates into a new column for the associated linear programming problem the algorithm has performed well on a set of convex nondifferentiable programming problems

regularity properties of solutions to the basic problem in the calculus of variations this paper concerns the basic problem in the calculus of variations minimize a functional j defined by j x j l t x t x t dt a over a class of arcs x whose values at a and b have been specified existence theory provides rather weak conditions under which the problem has a solution in the class of absolutely continuous arcs conditions which must be strengthened in order that the standard necessary conditions apply the question arises what necessary conditions hold merely under hypotheses of existence theory say the classical tonelli conditions it is shown that given a solution x there exists a relatively open subset q of a b of full measure on which x is locally lipschitz and satisfies a form of the euler lagrange equation the main theorem of which this is a corollary can also be used in conjunction with various classes of additional hypotheses to deduce the global smoothness of solutions three such classes are identified and results of bernstein tonelli and morrey are extended one of these classes is of a novel nature and its study implies the new result that when l is independent of t the solution has essentially bounded derivative

vibrational control of nonlinear systems vibrational controllability and transient behavior in the first part of this work 1 the criteria for the existence of stabilizing parametric oscillations have been derived in the present paper the problem of choosing the stabilizing vibrations is addressed and the properties of transient behavior of vibrationally controlled nonlinear systems are analyzed

specialized fast algorithms for iqc feasibility and optimization problems the conventional way to treat integral quadratic constraint iqc problems is to transform them into semi definite programs sdps sdps can then be solved using interior point methods which have been proven efficient this approach however is not always the most efficient since it introduces additional decision variables to the sdp and the additional decision variables sometimes largely increase the complexity of the problem in this paper we demonstrate how to solve iqc problems by other alternatives more specifically we consider two cutting plane algorithms we will show that in certain cases these cutting plane algorithms can solve iqc problems much faster than the conventional approach numerical examples as well as some explanations from the point of view of computational complexity are provided to support our point

vibrational control of a class of nonlinear systems by nonlinear multiplicative vibrations vibrational control is a nonclassical control principle which proposes a utilization of zero mean parametric excitation of a dynamical system for control purposes this paper extends nonlinear vibrational control theory developed in 8 9 to systems controlled by nonlinear multiplicative vibrations condition for partial vibrational stabilization with respect to a component of a steady state vector the choice of stabilizing vibrations and the transient motions are discussed for a certain practically important class of nonlinear vibrationally controlled systems the application of the results is demonstrated on the example of a catalytic reactor using a combination of numerical and analytical techniques

a cutting plane method for solving kyp sdps semidefinite programs originating from the kalman yakubovich popov lemma are convex optimization problems and there exist polynomial time algorithms that solve them however the number of variables is often very large making the computational time extremely long algorithms more efficient than general purpose solvers are thus needed to this end structure exploiting algorithms have been proposed based on the dual formulation in this paper a cutting plane algorithm is proposed in a comparison with a general purpose solver and a structure exploiting solver it is shown that the cutting plane based solver can handle optimization problems of much higher dimension

multiple cuts in the analytic center cutting plane method we analyze the multiple cut generation scheme in the analytic center cutting plane method we propose an optimal primal and dual updating direction when the cuts are central the direction is optimal in the sense that it maximizes the product of the new dual slacks and of the new primal variables within the trust regions defined by dikin s primal and dual ellipsoids the new primal and dual directions use the variance covariance matrix of the normals to the new cuts in the metric given by dikin s ellipsoid we prove that the recovery of a new analytic center from the optimal restoration direction can be done in o plog p 1 damped newton steps where p is the number of new cuts added by the oracle which may vary with the iteration the results and the proofs are independent of the specific scaling matrix primal dual or primal dual that is used in the computations the computation of the optimal direction uses newton s method applied to a self concordant function of p variables the convergence result of ye math programming 78 1997 pp 85 104 holds here also the algorithm stops after o frac bar p 2n 2 varepsilon 2 cutting planes have been generated where bar p is the maximum number of cuts generated at any given iteration

the approximation of higher order integrals of the calculus of variations and the lavrentiev phenomenon we prove the following approximation theorem given a function x a b to mathbb r n in the sobolev space bf w nu 1 1 nu geq 1 and epsilon 0 there exists a function x epsilon in bf w nu 1 infty such that begin gather int a b sum i 1 m l i x epsilon nu x epsilon nu 1 psi i t x epsilon x epsilon dotsc x epsilon nu int a b sum i 1 m l i x nu x nu 1 psi i t x x dotsc x nu epsilon begin aligned x epsilon a x a quad x epsilon b x b x epsilon a x a quad x epsilon b x b vdots x epsilon nu a x nu a quad x epsilon nu b x nu b end aligned end gather provided that for every i in 1 dotsc m l i psi i is continuous in a neighborhood of x l i is convex in its second variable and psi i evaluated along x has positive sign we discuss the optimality of our assumptions comparing them with an example of sarychev j dynam control systems 3 1997 pp 565 588 as a consequence we obtain the nonoccurrence of the lavrentiev phenomenon in particular the integral functional int a b l x nu x nu 1 does not exhibit the lavrentiev phenomenon for any given boundary values x a a x b b x a a x b b dotsc x nu a a nu x nu b b nu furthermore we prove the following necessary condition an action functional with lagrangian of the form sum i 1 m l i x nu x nu 1 psi i t x x dotsc x nu with nu geq 0 exhibiting the lavrentiev phenomenon takes the value infty in any neighborhood of a minimizer

existence and regularity in the small in the calculus of variations abstract a local existence theorem is proved for the basic problem in the calculus of variations that of minimizing l t x x dt over a class of functions x assuming given boundary conditions the lagrangian l is only assumed to be locally lipschitz and strictly convex in its x variable

on the computation of weighted analytic centers and dual ellipsoids with the projective algorithm the primal projective algorithm for linear programs with unknown optimal objective function value is extended to the case where one uses a weighted karmarkar potential function this potential is defined with respect to a strict lower bound to the optimum the minimization of this potential when the lower bound is kept fixed yields a primal and a dual feasible solution the dual solution is the weighted analytic center of a certain dual polytope finally one exhibits a pair of homothetic dual ellipsoids that extends results obtained by sonnevend todd ye freund and anstreicher

homogeneous analytic center cutting plane methods for convex problems and variational inequalities in this paper we consider a new analytic center cutting plane method in an extended space we prove the efficiency estimates for the general scheme and show that these results can be used in the analysis of a feasibility problem the variational inequality problem and the problem of constrained minimization our analysis is valid even for problems whose solution belongs to the boundary of the domain

complexity of some cutting plane methods that use analytic centers we consider cutting plane methods for minimizing a convex possibly nondifferentiable function subject to box constraints at each iteration accumulated subgradient cuts define a polytope that localizes the minimum the objective and its subgradient are evaluated at the analytic center of this polytope to produce one or two cuts that improve the localizing set we give complexity estimates for several variants of such methods our analysis is based on the works of goffin luo and ye

vibrational stabilization of nonlinear parabolic systems with neumann boundary conditions the conditions for the existence of the stabilizing vibrations for a class of distributed parameter systems governed by parabolic partial differential equations with neumann boundary conditions are derived the guidelines for the choice of the vibration parameters that ensure stabilization are given examples of vibrational stabilization of unstable systems by linear multiplicative and vector additive vibrations are given to support the theory

application of averaging method for integro differential equations to model reference adaptive control of parabolic systems abstract an averaging theorem for integro differential equations is applied to the convergence analysis of controller parameters of a model reference adaptive control algorithm for a class of parabolic partial differential equations with constant coefficients the stability of an adaptive control algorithm is proven as well

an interior point constraint generation algorithm for semi infinite optimization with health care application we propose an interior point constraint generation ipcg algorithm for semi infinite linear optimization silo and prove that the algorithm converges to an e solution of silo after a finite number of constraints is generated we derive a complexity bound on the number of newton steps needed to approach the updated center after adding multiple violated constraints and a complexity bound on the total number of constraints that is required for the overall algorithm to converge we implement our algorithm to solve the sector duration optimization problem arising in leksell gamma knife perfexion elekta stockholm sweden treatment planning a highly specialized treatment for brain tumors using real patient data provided by the department of radiation oncology at princess margaret hospital in toronto ontario canada we show that our algorithm can efficiently handle problems in real life health care applications by providing a quality treatment plan in a timely manner comparing our computational results with mosek a commercial software package we show that the ipcg algorithm outperforms the classical primal dual interior point methods on sector duration optimization problem arising in perfexion treatment planning we also compare our results with that of a projected gradient method in both cases we show that ipcg algorithm obtains a more accurate solution substantially faster

analysis of a cutting plane method that uses weighted analytic center and multiple cuts we consider the analytic center cutting plane or column generation algorithm for solving general convex problems defined by a separation oracle the oracle is called at an approximate analytic center of a polytope which contains the solution set and is given by the intersection of the linear inequalities previously generated from the oracle if the approximate center is not in the solution set separating hyperplanes will be placed through the approximate center and a new approximate analytic center will be found for the shrunken polytope in this paper we consider using approximate weighted analytic centers in the cutting plane method and show that the method with multiple cuts added in each step has a complexity of o eta m 2 epsilon 2 where eta is the maximum number of cuts that can be added in each step and m is the dimension of the problem

an analytic center cutting plane method for semidefinite feasibility problems semidefinite feasibility problems arise in many areas of operations research the abstract form of these problems can be described as finding a point in a nonempty bounded convex body g in the cone of symmetric positive semidefinite matrices assume that g is defined by an oracle which for any givenm xm symmetric positive semidefinite matrix y either confirms that y e g or returns a cut i e a symmetric matrixa such that g is in the half space y a y a y we study an analytic center cutting plane algorithm for this problem at each iteration the algorithm computes an approximate analytic center of a working set defined by the cutting plane system generated in the previous iterations if this approximate analytic center is a solution then the algorithm terminates otherwise the new cutting plane returned by the oracle is added into the system as the number of iterations increases the working set shrinks and the algorithm eventually finds a solution to the problem all iterates generated by the algorithm are positive definite matrices the algorithm has a worst case complexity of o m3 e 2 on the total number of cuts to be used where e is the maximum radius of a ball contained by g