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

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

on global hypoellipticity of degenerate elliptic operators we consider a class of degenerate elliptic operators on a torus and prove that global hypoellipticity is equivalent to an algebraic condition involving liouville vectors and simultaneous approximability for another class of operators we show that the zero order term may influence global hypoellipticity

newton polygon and distribution of integer points in sublevel sets let f x y be a polynomial in two variables of the form begin aligned f x y a 0y d a 1 x y d 1 cdots a d x end aligned where d is the degree of f for r 0 let begin aligned g f r x y in mathbb r 2 f x y le r end aligned we study the distribution of integer points in g f r in assuming that f satisfies the so called weekly degenerate condition for main edges of the complete newton polygon of f we show that

global properties of a class of overdetermined systems abstract we consider an overdetermined system of complex vector fields on the three dimensional torus which is naturally associated to a real analytic closed 1 form on the two dimensional torus by means of a detailed study of the geometry of level sets of a primitive of the pull back of the 1 form via the universal covering we prove that a necessary condition for the system to be globally solvable in the non exact case is that each connected component of the critical set has a point at which the local primitives of the 1 form are open maps when this condition is violated we also construct global solutions to the inhomogenous equations having analytic singularities

global and local analyticity for second order operators constructed from rigid vector fields on products of tori we prove global analytic hypoellipticity on a product of tori for partial differential operators which are constructed as rigid variable coefficient quadratic polynomials in real vector fields satisfying the h6rmander condition and where p satisfies a maximal estimate we also prove an analyticity result that is local in some variables and global in others for operators whose prototype is 0z 09x2 t1 with analytic a x a o 0 naturally but not identically zero the results because of the flexibility of the methods generalize recent work of cordaro and himonas in 4 and himonas in 8 which showed that certain operators known not to be locally analytic hypoelliptic those of baouendi and goulaouic 1 hanges and himonas 6 and christ 3 were globally analytic hypoelliptic on products of tori

on the global solvability in gevrey classes on the n dimensional torus abstract let p be a linear partial differential operator with coefficients in the gevrey class g s t n where t n is the n dimensional torus and s 1 we prove a necessary condition for the s global solvability of p on t n we also apply this result to give a complete characterization for the s global solvability for a class of formally self adjoint operators with nonconstant coefficients

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

first and second order analysis of nonlinear semidefinite programs in this paper we study nonlinear semidefinite programming problems convexity duality and first order optimality conditions for such problems are presented a second order analysis is also given second order necessary and sufficient optimality conditions are derived finally sensitivity analysis of such programs is discussed

complementarity and nondegeneracy in semidefinite programming primal and dual nondegeneracy conditions are defined for semidefinite programming given the existence of primal and dual solutions it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution the converses hold if strict complementarity is assumed primal and dual nondegeneracy assumptions do not imply strict complementarity as they do in lp the primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutions x and z this is in contrast with lp where nondegeneracy assumptions exactly determine the number of variables which are zero it is shown that primal and dual nondegeneracy and strict complementarity all hold generically numerical experiments suggest probability distributions for the ranks of x and z which are consistent with the nondegeneracy conditions

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

clarke subgradients for directionally lipschitzian stratifiable functions using a geometric argument we show that under a reasonable continuity condition the clarke subdifferential of a semi algebraic or more generally stratifiable directionally lipschitzian function admits a simple form the normal cone to the domain and limits of gradients generate the entire clarke subdifferential the characterization formula we obtain unifies various apparently disparate results that have appeared in the literature our techniques also yield a simplified proof that closed semialgebraic functions on rn have a limiting subdifferential graph of uniform local dimension n

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

regularity of a class of sublaplacians on the 3 dimensional torus abstract in this paper we present a necessary and sufficient condition for a family of sums of squares operators to be globally hypoelliptic on a torus this condition says that either a diophantine condition is satisfied or there exists a point of finite type also we describe the analytic and gevrey versions of this result the proof is based on l 2 estimates and microlocal analysis

prescribing analytic singularities for solutions of a class of vector fields on the torus we consider the operator l t a t ib t x acting on distributions on the two torus t 2 where and b are real valued real analytic functions defined on the unit circle t 1 we prove among other things that when b changes sign given any subset s of the set of the local extrema of the local primitives of b there exists a singular solution of l such that the t projection of its analytic singular support is furthermore for any t l and any closed subset f of t 1 x there exists u d t 2 such that lu c t 2 and sing supp a u x f we also provide a microlocal result concerning the truce of u at t

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

global solvability and simultaneously approximable vectors abstract we consider a class of sum of squares operators on a torus and we prove that global solvability is equivalent to an algebraic condition involving simultaneously approximable vectors

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

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

global s solvability and global s hypoellipticity for certain perturbations of zero order of systems of constant real vector fields abstract the main purpose of this paper is to show in the two dimensional torus a necessary and sufficient condition in order to certain perturbations of zero order of a system of constant real vector fields to be globally s solvable we are also interested in studying its global s hypoellipticity we present connections between these global concepts and a priori estimates we also present two applications of our results for systems of operators with variable coefficients

generic optimality conditions for semialgebraic convex programs we consider linear optimization over a nonempty convex semialgebraic feasible region f semidefinite programming is an example if f is compact then for almost every linear objective there is a unique optimal solution lying on a unique active manifold around which f is partly smooth and the second order sufficient conditions hold perturbing the objective results in smooth variation of the optimal solution the active manifold consists locally of these perturbed optimal solutions it is independent of the representation of f and is eventually identified by a variety of iterative algorithms such as proximal and projected gradient schemes these results extend to unbounded sets f

lagrange multipliers and optimality lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write first order optimality conditions formally as a system of equations modern applications with their emphasis on numerical methods and more complicated side conditions than equations have demanded deeper understanding of the concept and how it fits into a larger theoretical picture a major line of research has been the nonsmooth geometry of one sided tangent and normal vectors to the set of points satisfying the given constraints another has been the game theoretic role of multiplier vectors as solutions to a dual problem interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows black and white constraints to be replaced by penalty expressions this paper traces su

global hypoellipticity and global solvability in gevrey classes on the n dimensional torus let p be a linear partial differential operator with coefficients in the gevrey class gs tn where tn is the n dimensional torus and s 1 we prove that if p is s globally hypoelliptic in tn then its transposed operator tp is s globally solvable in tn thus extending to the gevrey classes the well known analogous result in the corresponding c class

conditions ne cessaires d hypoanalyticite pour des ope rateurs invariantsa gauche homoge nes sur un groupe nilpotent gradue

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

remarks about global analytic hypoellipticity we present a characterization of the operators l 8 ot a t ib t x which are globally analytic hypoelliptic on the torus we give information about the global analytic hypoellipticity of certain overdetermined systems and of sums of squares

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

manifold identification in dual averaging for regularized stochastic online learning iterative methods that calculate their steps from approximate subgradient directions have proved to be useful for stochastic learning problems over large and streaming data sets when the objective consists of a loss function plus a nonsmooth regularization term the solution often lies on a low dimensional manifold of parameter space along which the regularizer is smooth when an l1 regularizer is used to induce sparsity in the solution for example this manifold is defined by the set of nonzero components of the parameter vector this paper shows that a regularized dual averaging algorithm can identify this manifold with high probability before reaching the solution this observation motivates an algorithmic strategy in which once an iterate is suspected of lying on an optimal or near optimal manifold we switch to a local phase that searches in this manifold thus converging rapidly to a near optimal point computational results are presented to verify the identification property and to illustrate the effectiveness of this approach

closedness of the range for vector fields on the torus abstract we consider vector fields of the form l t a x x on the 2 torus and study the closedness of the range of l in spaces of functions and distributions when a vanishes somewhere but not everywhere we show that the range is closed if and only if all zeros of a are of finite order in the most interesting case one obtains a finiteness condition on the speed of accumulation of orbits similar to what happens in the known tube case

on finite convergence and constraint identification of subgradient projection methods this paper deals with a continuous time subgradient projection algorithm shown to generate trajectories that accumulate to the solution set under a strong convexity assumption we show that convergence is exponential in norm a sharpness condition yields convergence in finite time and the necessary lapse is estimated invoking a constraint qualification and a non degeneracy assumption we demonstrate that optimally active constraints are identified in finite time

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

quadratic growth and critical point stability of semi algebraic functions we show that quadratic growth of a semi algebraic function is equivalent to strong metric subregularity of the subdifferential a kind of stability of generalized critical points in contrast this equivalence can easily fail outside of the semi algebraic setting as a consequence we derive necessary conditions and sufficient conditions for optimality in subdifferential terms