relative entropy optimization and its applications in this expository article we study optimization problems specified via linear and relative entropy inequalities such relative entropy programs reps are convex optimization problems as the relative entropy function is jointly convex with respect to both its arguments prominent families of convex programs such as geometric programs gps second order cone programs and entropy maximization problems are special cases of reps although reps are more general than these classes of problems we provide solutions based on reps to a range of problems such as permanent maximization robust optimization formulations of gps and hitting time estimation in dynamical systems we survey previous approaches to some of these problems and the limitations of those methods and we highlight the more powerful generalizations afforded by reps we conclude with a discussion of quantum analogs of the relative entropy function including a review of the similarities and distinctions with respect to the classical case we also describe a stylized application of quantum relative entropy optimization that exploits the joint convexity of the quantum relative entropy function

a polynomial time approximation algorithm for the permanent of a matrix with nonnegative entries we present a polynomial time randomized algorithm for estimating the permanent of an arbitrary n n matrix with nonnegative entries this algorithm technically a fully polynomial randomized approximation scheme computes an approximation that is with high probability within arbitrarily small specified relative error of the true value of the permanent

lifts of convex sets and cone factorizations in this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone such a representation or lift of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices we show that the existence of a lift of a convex set to a cone is equivalent to the existence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual this generalizes a theorem of yannakakis that established a connection between polyhedral lifts of a polytope and nonnegative factorizations of its slack matrix symmetric lifts of convex sets can also be characterized similarly when the cones live in a family our results lead to the definition of the rank of a convex set with respect to this family we present results about this rank in the context of cones of positive semidefinite matrices our methods provide new tools for understanding cone lifts of convex sets

sufficient and necessary conditions for semidefinite representability of convex hulls and sets a set s subseteq mathbb r n is called semidefinite programming sdp representable or semidefinite representable if s equals the projection of a set in higher dimensional space which is describable by some linear matrix inequality lmi clearly if s is sdp representable then s must be convex and semialgebraic it is describable by conjunctions and disjunctions of polynomial equalities or inequalities this paper proves sufficient conditions and necessary conditions for sdp representability of convex sets and convex hulls by proposing a new approach to constructing sdp representations the contributions of this paper are i for bounded sdp representable sets w 1 dots w m we give an explicit construction of an sdp representation for mathrm conv cup k 1 mw k this provides a technique for building global sdp representations from the local ones ii for the sdp representability of a compact convex semialgebraic set s we prove sufficient the boundary partial s is nonsingular and positively curved while necessary is partial s has nonnegative curvature at each nonsingular point in terms of defining polynomials for s nonsingular boundary amounts to them having nonvanishing gradient at each point on partial s and the curvature condition can be expressed as their strict versus nonstrict quasi concavity at those points on partial s where they vanish the gaps between them are partial s having or not having singular points either of the gradient or of the curvature s positivity a sufficient condition bypassing the gaps is when some defining polynomials of s satisfy an algebraic condition called sos concavity iii for the sdp representability of the convex hull of a compact nonconvex semialgebraic set t we find that the critical object is partial c t the maximum subset of partial t contained in partial mathrm conv t we prove sufficient for sdp representability partial c t is nonsingular and positively curved and necessary is partial c t has nonnegative curvature at nonsingular points the gaps between our sufficient and necessary conditions are similar to case ii the positive definite lagrange hessian pdlh condition which meshes well with constructions is also discussed

capacities of quantum channels and how to find them we survey what is known about the information transmitting capacities of quantum channels and give a proposal for how to calculate some of these capacities using linear programming

periodic solutions of the ginzburg landau equation abstract spatially periodic solutions to the ginzburg landau equation are considered in particular we obtain criteria for primary and secondary bifurcation limit cycle solutions nonlinear dispersion relations relating spatial and temporal frequencies only relatively simple tools appear in the treatment and as a result a wide range of parameter cases are considered finally we briefly treat the case of spatial bifurcations

symmetry classes of variable coefficient nonlinear schrodinger equations a variable coefficient nonlinear schrodinger vcnls equation involving three arbitrary complex functions of space time in 1 1 dimensions is analysed from the point of view of its symmetries all equations of the type studied having non trivial lie point symmetry groups g are identified the symmetry group is shown to be at most five dimensional and only when the equation is equivalent to the nls equation itself or to a rather special complex ginzburg landau equation lie point transformations are used to obtain solutions of specific vcnls equations that should be of interest in nonlinear optics or other branches of physics

the capacity of the quantum depolarizing channel the information carrying capacity of the d dimensional depolarizing channel is computed it is shown that this capacity can be achieved by encoding messages as products of pure states belonging to an orthonormal basis of the state space and using measurements which are products of projections onto this same orthonormal basis in other words neither entangled signal states nor entangled measurements give any advantage for information capacity the result follows from an additivity theorem for the product channel spl delta spl ominus spl psi where spl delta is the depolarizing channel and spl psi is a completely arbitrary channel we establish the amosov holevo werner see probl inform transm vol 36 p 305 313 2000 p norm conjecture for this product channel for all p spl ges 1 and deduce from this the additivity of the minimal entropy and of the holevo quantity spl chi sup

inequalities for quantum entropy a review with conditions for equality this article presents self contained proofs of the strong subadditivity inequality for von neumann s quantum entropy s and some related inequalities for the quantum relative entropy most notably its convexity and its monotonicity under stochastic maps moreover the approach presented here which is based on klein s inequality and lieb s theorem that the function a tr ek log a is concave allows one to obtain conditions for equality in the case of strong subadditivity which states that s 123 s 2 s 12 s 23 where the subscripts denote subsystems of a composite system equality holds if and only if log 123 log 12 log 2 log 23 using the fact that the holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy we show that equality can be attained for that bound only when the states in the ensemble commute the article concludes with an appendix giving a short description of epstein s elegant proof of lieb s the

periodic solutions and stability for a singularly perturbed linear delay differential equation abstract we describe the bifurcation hypersurfaces for periodic solutions of a singularly perturbed linear differential difference equation in the space of the coefficients of the equation for low dimension we show that the locus of stability of that equation approaches the locus of stability of a limit difference equation

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

homoclinic explosions in the complex ginzburg landau equation a bifurcation sequence of the complex ginzburg landau cgl equation in an even subspace of spatially periodic solutions is analyzed with local three dimensional bifurcation theory near homoclinic orbits bifurcation to chaos occurs via homoclinic explosions similar to bifurcation to chaos in the lorenz model and a similar global bifurcation picture is developed an analysis via one dimensional maps is developed to predict attractors it is shown that the two mode galerkin truncation can mimic some bifurcations of the full cgl equationdeceptively and also that the truncation fails to mimic many features of full pde precisely because the truncation lacks many of the same homoclinic orbits

the capacity of the quantum channel with general signal states it is shown that the capacity of a classical quantum channel with arbitrary possibly mixed states equals the maximum of the entropy bound with respect to all a priori distributions this completes the recent result of hausladen jozsa schumacher westmoreland and wootters 1996 who proved the equality for the pure state channel

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

on polyhedral approximations of the second order cone we demonstrate that a conic quadratic problem is polynomially reducible to linear programming we demonstrate this by constructing for every an lp program explicitly given in terms of e and the data of cqp with the following properties i the number dimx dimu of variables and the number dimp of constraints in lp do not exceed ii every feasible solutionx to cqp can be extended to a feasible solution x u to lp iii if x u is feasible for lp thenx satisfies the e relaxed constraints of cqp namely

primal dual potential reduction algorithm for symmetric programming problems with nonlinear objective functions abstract we consider a primal dual potential reduction algorithm for nonlinear convex optimization problems over symmetric cones the same complexity estimates as in the case of the linear objective function are obtained provided a certain nonlinear system of equations can be solved with a given accuracy this generalizes the result of k kortanek f potra and y ye 7 we further introduce a generalized nesterov todd direction and show how it can be used to achieve a required accuracy by solving the linearization of above mentioned nonlinear system for a class of nonlinear convex functions satisfying scaling lipschitz condition this result is a far reaching generalization of results of f potra y ye and j zhu 8 9 finally we show that a class of functions which contains quantum entropy function satisfies scaling lipschitz condition

chaotic dynamics of coherent structures abstract a variety of chaotic flows evolving in relatively high dimensional spaces are considered it is shown through the use of an optimal choice of basis functions which are consequence of the karhunen loeve procedure that an accurate description can be given in a relatively low dimensional space particular examples of this procedure which are presented are the ginzburg landau equation turbulent convection in an unbounded domain and turbulent convection in a bounded domain

geometric programming duals of channel capacity and rate distortion we show that the lagrange dual problems of the channel capacity problem with input cost and the rate distortion problem are simple geometric programs upper bounds on channel capacity and lower bounds on rate distortion can be efficiently generated from their duals for channel capacity the geometric programming dual characterization is shown to be equivalent to the minmax kullback leibler kl characterization in csiszar et al 1981 for rate distortion the geometric programming dual is extended to rate distortion with two sided state information a duality by mapping is then given between the lagrange dual problems of channel capacity with input cost and rate distortion which resolves several apparent asymmetries between their primal problems in the familiar form of mutual information optimization problems both the primal and dual problems can be interpreted in a common framework of free energy optimization from statistical physics

low dimensional models of the ginzburg landau equation low dimensional numerical approximations for two boundary value problems of the complex ginzburg landau equation in a chaotic regime are constructed spatial structures called principal interaction patterns are extracted from the system according to a nonlinear variational principle and used as basis functions in a galerkin approximation the dynamical description in terms of principal interaction patterns requires fewer modes than more conventional approaches using fourier modes or karhunen loeve modes as basis functions

low dimensional dynamics for the complex ginzburg landau equation abstract a method is presented which results in low dimensional dynamical systems for situations in which low dimensional attractors are known to exist the method is based on the use of the karhunen loeve procedure for the determination of an optimal basis and the subsequent use of the galerkin procedure to generate the dynamical system the method is applied to two problems for the ginzburg landau equation for which large databases have been obtained in each instance a dynamical system is generated which has roughly twice the number of degrees of freedom as the hausdorff dimension of the exact case it is also demonstrated that the approximations are robust in that they are accurate over a wide range of parameter space

low rank kernel learning with bregman matrix divergences in this paper we study low rank matrix nearness problems with a focus on learning low rank positive semidefinite kernel matrices for machine learning applications we propose efficient algorithms that scale linearly in the number of data points and quadratically in the rank of the input matrix existing algorithms for learning kernel matrices often scale poorly with running times that are cubic in the number of data points we employ bregman matrix divergences as the measures of nearness these divergences are natural for learning low rank kernels since they preserve rank as well as positive semidefiniteness special cases of our framework yield faster algorithms for various existing learning problems and experimental results demonstrate that our algorithms can effectively learn both low rank and full rank kernel matrices

dimension of the attractors associated to the ginzburg landau partial differential equation abstract we study the long time behavior of solutions to the ginzburg landau partial differential equation it is shown that a finite dimensional attractor captures all the solutions an upper bound on this dimension is given in terms of physical quantities by estimating the lyapunov exponents on the trajectories finally using the well known side band instabilities of an exact time dependent solution stokes solution we derive lower bounds on the dimension of the universal attractor moreover the lower and upper bounds agree

relative entropy relaxations for signomial optimization signomial programs sps are optimization problems specified in terms of signomials which are weighted sums of exponentials composed with linear functionals of a decision variable sps are nonconvex optimization problems in general and families of np hard problems can be reduced to sps in this paper we describe a hierarchy of convex relaxations to obtain successively tighter lower bounds of the optimal value of sps this sequence of lower bounds is computed by solving increasingly larger sized relative entropy optimization problems which are convex programs specified in terms of linear and relative entropy functions our approach relies crucially on the observation that the relative entropy function by virtue of its joint convexity with respect to both arguments provides a convex parametrization of certain sets of globally nonnegative signomials with efficiently computable nonnegativity certificates via the arithmetic geometric mean inequality by appealing to representation theorems from real algeb

the asymptotic stability of xn 1 axn bxn k 0 we obtain necessary and sufficient conditions for the asymptotic stability of the difference equation xn 1 axn bxn k 0 n 0 1 2 where the coefficients a and b are real numbers and the delay k is a nonnegative integer

efficient optimization of the quantum relative entropy many quantum information measures can be written as an optimization of the quantum relative entropy between sets of states for example the relative entropy of entanglement of a state is the minimum relative entropy to the set of separable states the various capacities of quantum channels can also be written in this way we propose a unified framework to numerically compute these quantities using off the shelf semidefinite programming solvers exploiting the approximation method proposed in fawzi saunderson and parrilo 2017 arxiv 1705 00812 as a notable application this method allows us to provide numerical counterexamples for a proposed lower bound on the quantum conditional mutual information in terms of the relative entropy of recovery

two boundary value problems for the ginzburg landau equation abstract two boundary value problems for the ginzburg landau equation are considered extensive numerical calculations have been performed in each case including bifurcation histories spectral analysis poincare sections and hausdorff dimension estimates the approach to the inviscid limit is given detailed treatment in this case universal behavior has been found to exist arguments are presented to account for this behavior

lieb s concavity theorem matrix geometric means and semidefinite optimization abstract a famous result of lieb establishes that the map a b tr k a 1 t k b t is jointly concave in the pair a b of positive definite matrices where k is a fixed matrix and t 0 1 in this paper we show that lieb s function admits an explicit semidefinite programming formulation for any rational t 0 1 our construction makes use of a semidefinite formulation of weighted matrix geometric means we provide an implementation of our constructions in matlab

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow