the adaptive convexification algorithm for semi infinite programming with arbitrary index sets a numerical solution method for semi infinite optimization problems with arbitrary not necessarily box shaped index sets is presented following the ideas of floudas and stein siam j optim 18 1187 1208 2007 convex relaxations of the lower level problem are adaptively constructed and then reformulated as mathematical programs with complementarity constraints and solved although the index set is arbitrary this approximation produces feasible iterates for the original problem the convex relaxations and needed parameters are constructed with ideas of the bb method of global optimization and interval methods it is shown that after finitely many steps an epsilon stationary point of the original semi infinite problem is reached a numerical example illustrates the performance of the proposed method

a new general purpose algorithm for mixed integer bilevel linear programs bilevel optimization problems are very challenging optimization models arising in many important practical contexts including pricing mechanisms in the energy sector airline and telecommunication industry transportation networks critical infrastructure defense and machine learning in this paper we consider bilevel programs with continuous and discrete variables at both levels with linear objectives and constraints continuous upper level variables if any must not appear in the lower level problem we propose a general purpose branch and cut exact solution method based on several new classes of valid inequalities which also exploits a very effective bilevel specific preprocessing procedure an extensive computational study is presented to evaluate the performance of various solution methods on a common testbed of more than 800 instances from the literature and 60 randomly generated instances our new algorithm consistently outperforms often by a large margin alternative state of the art metho

the word problem for ω terms over da in this paper we solve the word problem for terms over da we extend to da the ideas used by almeida and zeitoun to solve the analogous problem for the pseudovariety r applying also a representation by automata of implicit operations on da which was recently obtained by the author considering certain types of factors of an implicit operation on da we can prove that a pseudoword on da is an term if and only if the associated minimal da automaton is finite finally we complete the result by effectively computing in polynomial time the minimal da automaton associated to an term

exact penalization of mathematical programs with equilibrium constraints we study theoretical and computational aspects of an exact penalization approach to mathematical programs with equilibrium constraints mpecs in the first part we prove that a mangasarian fromovitz type condition ensures the existence of a stable local error bound at the root of a real valued nonnegative piecewise smooth function a specification to nonsmooth formulations of equilibrium constraints e g complementarity conditions or normal equations provides conditions which guarantee the existence of a nonsmooth exact penalty function for mpecs in the second part we study a trust region minimization method for a class of composite nonsmooth functions which comprises exact penalty functions arising from mpecs we prove a global convergence result for the general method and incorporate a penalty update rule a further specification results in an sqp trust region method for mpecs based on an ell 1 penalty function

nonlinear robust optimization via sequential convex bilevel programming in this paper we present a novel sequential convex bilevel programming algorithm for the numerical solution of structured nonlinear min max problems which arise in the context of semi infinite programming here our main motivation are nonlinear inequality constrained robust optimization problems in the first part of the paper we propose a conservative approximation strategy for such nonlinear and non convex robust optimization problems under the assumption that an upper bound for the curvature of the inequality constraints with respect to the uncertainty is given we show how to formulate a lower level concave min max problem which approximates the robust counterpart in a conservative way this approximation turns out to be exact in some relevant special cases and can be proven to be less conservative than existing approximation techniques that are based on linearization with respect to the uncertainties in the second part of the paper we review existing theory on optimality conditions for nonlinear lower level concave min max problems which arise in the context of semi infinite programming regarding the optimality conditions for the concave lower level maximization problems as a constraint of the upper level minimization problem we end up with a structured mathematical program with complementarity constraints mpcc the special hierarchical structure of this mpcc can be exploited in a novel sequential convex bilevel programming algorithm we discuss the surprisingly strong global and locally quadratic convergence properties of this method which can in this form neither be obtained with existing sqp methods nor with interior point relaxation techniques for general mpccs finally we discuss the application fields and implementation details of the new method and demonstrate the performance with a numerical example

algebraic cycles and higher k theory here g x is the grothendieck group of coherent sheaves on x 13 gr refers to the graded group defined by the y filtration on g x cf kratzer 14 soult 20 and ch x is the chow group of codimension i algebraic cycles defined by fulton 9 the left hand isomorphism is a formal consequence of the existence of a i structure on g x while the existence of r is the central theme of the b f m rr theorem the main purpose of this paper is to define a theory of higher chow groups ch x n n 2 0 so as to obtain isomorphisms

semi infinite programming theory methods and applications starting from a number of motivating and abundant applications in 2 including control of robots eigenvalue computations mechanical stress of materials and statistical design the authors describe a class of optimization problems which are referred to as semi infinite because their constraints bound functions of a finite number of variables on a whole region in 3 5 first and second order optimality conditions are derived for general nonlinear problems as well as a procedure for reducing the problem locally to one with only finitely many constraints another main effort for achieving simplification is through duality in 6 there algebraic properties of finite linear programming are brought to bear on duality theory in semi infinite programming section 7 treats numerical methods based on either discretization or local reduction with the emphasis on the design of superlinearly convergent sqp type methods taking this differentiable point of view this paper can be considered to be complementar

triangulated categories of relative 1 motives abstract the category da 1 s of relative cohomological 1 motives is the localizing subcategory of the triangulated category da s da e t s q of relative voevodsky motives with rational coefficients over a scheme s which is generated by cohomological motives of curves over s we construct and study a candidate for the standard motivic t structure on da 1 s for s noetherian finite dimensional and excellent we show this t structure is non degenerate and relate its heart mm 1 s with deligne 1 motives over s in particular when s is regular the category of deligne 1 motives embeds in mm 1 s fully faithfully we also study the inclusion of da 1 s into the larger category da coh s of relative cohomological motives on s and prove that its right adjoint 1 preserves compact objects

l algebre de hopf et le groupe de galois motiviques d un corps de caracteristique nulle ii this is the second article of a series of two aiming at constructing and studying motivic galois groups in the context of triangulated motives these motivic galois groups were constructed in the first article and their algebras of regular functions were described concretely in terms of differential forms or algebraic cycles in the present article we gather important complements to the first one in the first part of this article we describe the link between the motivic galois group and the usual galois group of a subfield of in the second part we develop the basis of a theory of ramification for the motivic galois groups by constructing motivic versions of the decomposition and inertia groups associated to a geometric valuation we also introduce the relative motivic galois group of an extension k k which measures the difference between the motivic galois groups of k and k the latter is more accessible than its absolute counterpart in effect we show that it is a quotient of the pro algebraic completion of the topological fundamental pro group of a the pro variety homk k at least when the extension k k is of finite type

motivic periods and grothendieck arithmetic invariants abstract we construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers for the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period regulator is surjective showing that a suitable betti de rham realization of 1 motives is fully faithful we can verify this period conjecture in several cases the divisibility properties of motivic cohomology imply that our conjecture is a neat generalization of the classical grothendieck period conjecture for algebraic cycles on smooth and proper schemes these divisibility properties are treated in an appendix by b kahn extending previous work of bloch and colliot thelene raskind

on generalized semi infinite optimization and bilevel optimization the paper studies the connections and differences between bilevel problems bl and generalized semi infinite problems gsip under natural assumptions gsip can be seen as a special case of a bl we consider the so called reduction approach for bl and gsip leading to optimality conditions and newton type methods for solving the problems we show by a structural analysis that for gsip problems the regularity assumptions for the reduction approach can be expected to hold generically at a solution but for general bl problems not the genericity behavior of bl and gsip is in particular studied for linear problems

the κ word problem over drh abstract profinite techniques are explored in order to prove decidability of a word problem over a family of pseudovarieties of semigroups which is parameterized by pseudovarieties of groups let be the signature that naturally generalizes the usual signature on groups it consists of the multiplication and of the 1 power given a pseudovariety of groups h we denote by drh the pseudovariety all finite semigroups whose regular r classes lie in h we prove that the word problem for terms is decidable over drh provided it is decidable over h in general the word problem for terms is said to be decidable over a pseudovariety v if it is decidable whether two terms define the same element in every semigroup of v further we present a canonical form for elements in the free semigroup over drh based on the knowledge of a canonical form for elements in the free semigroup over h this extends work of almeida and zeitoun on the pseudovariety of all finite r trivial semigroups

local optimization of dynamic programs with guaranteed satisfaction of path constraints an algorithm is proposed for locating a feasible point satisfying the kkt conditions to a specified tolerance of feasible inequality path constrained dynamic programs pcdp within a finite number of iterations the algorithm is based on iteratively approximating the pcdp by restricting the right hand side of the path constraints and enforcing the path constraints at finitely many time points the main contribution of this article is an adaptation of the semi infinite program sip algorithm proposed in mitsos 2011 to pcdp it is proved that the algorithm terminates finitely with a guaranteed feasible point which satisfies the first order kkt conditions of the pcdp to a specified tolerance the main assumptions are i availability of a nonlinear program nlp local solver that generates a kkt point of the constructed approximation to pcdp at each iteration if this problem is indeed feasible ii existence of a slater point of the pcdp that also satisfies the first order kkt conditions of the pcdp to a specified tolerance iii all kkt multipliers are nonnegative and uniformly bounded with respect to all iterations the performance of the algorithm is analyzed through two numerical case studies

computing a global optimal solution to a design centering problem in this paper we present a method for solving a special three dimensional design centering problem arising in diamond manufacturing find inside a given not necessarily convex polyhedral rough stone the largest diamond of prescribed shape and orientation this problem can be formulated as the one of finding a global maximum of a difference of two convex functions over 3 and can be solved efficiently by using a global optimization algorithm provided that the objective function of the maximization problem can be easily evaluated here we prove that with the information available on the rough stone and on the reference diamond evaluating the objective function at a pointx amounts to computing the distance with respect to a minkowski gauge fromx to a finite number of planes we propose a method for finding these planes and we report some numerical results

an automata theoretic approach to the word problem for ω terms over r this paper studies the pseudovariety r of all finite r trivial semigroups we give a representation of pseudowords over r by infinite trees called r trees then we show that a pseudoword is an w term if and only if its associated tree is regular i e it can be folded into a finite graph or equivalently if the w term has a finite number of tails we give a linear algorithm to compute a compact representation of the r tree for w terms which yields a linear solution of the word problem for w terms over r we finally exhibit a basis for the w variety generated by r and we show that there is no finite basis several results can be compared to recent work of bloom and choffrut on long words

mccormick based relaxations of algorithms theory and implementation for the global optimization of a wide class of algorithms is presented via convex affine relaxations the basis for the proposed relaxations is the systematic construction of subgradients for the convex relaxations of factorable functions by mccormick math prog 10 1976 pp 147 175 similar to the convex relaxation the subgradient propagation relies on the recursive application of a few rules namely the calculation of subgradients for addition multiplication and composition operations subgradients at interior points can be calculated for any factorable function for which a mccormick relaxation exists provided that subgradients are known for the relaxations of the univariate intrinsic functions for boundary points additional assumptions are necessary an automated implementation based on operator overloading is presented and the calculation of bounds based on affine relaxation is demonstrated for illustrative examples two numerical examples for the global optimization of algorithms are presented in both examples a parameter estimation problem with embedded differential equations is considered the solution of the differential equations is approximated by algorithms with a fixed number of iterations

on the mathematical foundations of nondifferentiable optimization in engineering design it is shown by example that a large class of engineering design problems can be transcribed into the form of a canonical optimization problem with inequality constraints involving mar functions such problems are commonly referred to as semi infinite optimization problems the bulk of this paper is devoted to the development of a mathematical theory for the construction of first order nondifferentiable optimization algorithms related to phase i phase ii methods of feasible directions which solve these semi infinite optimization problems the applicability of the theory is illustrated with examples that are relevant to engineering design

the adaptive convexification algorithm a feasible point method for semi infinite programming we present a new numerical solution method for semi infinite optimization problems its main idea is to adaptively construct convex relaxations of the lower level problem replace the relaxed lower level problems equivalently by their karush kuhn tucker conditions and solve the resulting mathematical programs with complementarity constraints this approximation produces feasible iterates for the original problem the convex relaxations are constructed with ideas from the alpha bb method of global optimization the necessary upper bounds for second derivatives of functions on box domains can be determined using the techniques of interval arithmetic where our algorithm already works if only one such bound is available for the problem we show convergence of stationary points of the approximating problems to a stationary point of the original semi infinite problem within arbitrarily given tolerances numerical examples from chebyshev approximation and design centering illustrate the performance of the method

how to solve a semi infinite optimization problem abstract after an introduction to main ideas of semi infinite optimization this article surveys recent developments in theory and numerical methods for standard and generalized semi infinite optimization problems particular attention is paid to connections with mathematical programs with complementarity constraints lower level wolfe duality semi smooth approaches as well as branch and bound techniques in adaptive convexification procedures a section on recent genericity results includes a discussion of the symmetry effect in generalized semi infinite optimization

guaranteed satisfaction of inequality state constraints in pde constrained optimization abstract optimal control problems with inequality path constraints ipcs are present in several engineering problems described by partial differential equations pde we propose an algorithm to solve pde constrained dynamic optimization problems with guaranteed satisfaction of ipcs the algorithm is based on a solution of a sequence of approximated dynamic optimization problems following the strategy of fu et al automatica 2015 for the approximation the path constraint is imposed only on a set of discrete points and is thus relaxed at the same time it is restricted by an adaptive back off and by the error bound of the pde solution the approximation is iteratively improved at iterations without violation the restriction is reduced at iterations with violation the point of maximal violation is added to the discrete set under mild assumptions we prove finite termination of the algorithm we test the algorithm on an optimal grade change of a tubular reactor

rigidity in motivic homotopy theory we show that extensions of algebraically closed fields induce full and faithful functors between the respective motivic stable homotopy categories with finite coefficients

solving semi infinite optimization problems with interior point techniques we introduce a new numerical solution method for semi infinite optimization problems with convex lower level problems the method is based on a reformulation of the semi infinite problem as a stackelberg game and the use of regularized nonlinear complementarity problem functions this approach leads to central path conditions for the lower level problems where for a given path parameter a smooth nonlinear finite optimization problem has to be solved the solution of the semi infinite optimization problem then amounts to driving the path parameter to zero we show convergence properties of the method and give a number of numerical examples from design centering and from robust optimization where actually so called generalized semi infinite optimization problems are solved the presented method is easy to implement and in our examples it works well for dimensions of the semi infinite index set at least up to 150

mixed weil cohomologies abstract we define for a regular scheme s and a given field of characteristic zero k the notion of k linear mixed weil cohomology on smooth s schemes by a simple set of properties mainly nisnevich descent homotopy invariance stability which means that the cohomology of g m behaves correctly and kunneth formula we prove that any mixed weil cohomology defined on smooth s schemes induces a symmetric monoidal realization of some suitable triangulated category of motives over s to the derived category of the field k this implies a finiteness theorem and a poincare duality theorem for such a cohomology with respect to smooth and projective s schemes which can be extended to smooth s schemes when s is the spectrum of a perfect field this formalism also provides a convenient tool to understand the comparison of such cohomology theories

global solution of semi infinite programs optimization problems involving a finite number of decision variables and an infinite number of constraints are referred to as semi infinite programs sips existing numerical methods for solving nonlinear sips make strong assumptions on the properties of the feasible set e g convexity and or regularity or solve a discretized approximation which only guarantees a lower bound to the true solution value of the sip here a general deterministic algorithm for solving semi infinite programs to guaranteed global optimality is presented a branch and bound bb the lower bounding problem is formulated as a convex relaxation of a discretized approximation to the sip the sip b b algorithm is shown to converge finitely to optimality when the subdivision and discretization procedures used to formulate the node subproblems are known to retain certain convergence characteristics other than the properties assumed by globally convergent b b methods for finitely constrained nlps this sip algorithm makes only one additional assumption for every minimizer x of the sip there exists a sequence of slater points xn for which cf section 5 4 numerical results for test problems in the sip literature are presented the exclusion test and a modified upper bounding problem are also investigated as heuristic approaches for alleviating the computational cost of solving a nonlinear sip to guaranteed global optimality

a standard branch and bound approach for nonlinear semi infinite problems this paper considers nonlinear semi infinite problems which contain at least one semi infinite constraint sic the standard branch and bound algorithm is adapted to such problems by extending usual upper and lower bounding techniques for nonlinear inequality constraints to sics this is achieved by defining the interval evaluation of parametrized functions and their generalized gradients by also adapting numerical constraint programming techniques to quantified inequalities and by introducing linear relaxations and restrictions for sics the overall efficiency of our algorithm is demonstrated on a standard set of benchmarks from the literature in comparison with the best state of the art alternative

pessimistic bilevel optimization we study a variant of the pessimistic bilevel optimization problem which comprises constraints that must be satisfied for any optimal solution of a subordinate lower level optimization problem we present conditions that guarantee the existence of optimal solutions in such a problem and we characterize the computational complexity of various subclasses of the problem we then focus on problem instances that may lack convexity but that satisfy a certain independence property we develop convergent approximations for these instances and we derive an iterative solution scheme that is reminiscent of the discretization techniques used in semi infinite programming we also present a computational study that illustrates the numerical behavior of our algorithm on standard benchmark instances

feasible method for semi infinite programs a new numerical method is presented for semi infinite optimization problems which guarantees that each iterate is feasible for the original problem the basic idea is to construct concave relaxations of the lower level problem to compute the optimal values of the relaxation problems explicitly and to solve the resulting approximate problems with finitely many constraints the concave relaxations are constructed by replacing the objective function of the lower level problem by its concave upper bound functions under mild conditions we prove that every accumulation point of the solutions of the approximate problems is an optimal solution of the original problem an adaptive subdivision algorithm is proposed to solve semi infinite optimization problems it is shown that the karush kuhn tucker points of the approximate problems converge to a karush kuhn tucker point of the original problem within arbitrarily given tolerances numerical experiments show that our algorithm is much faster than the existi

relaxation based bounds for semi infinite programs finite formulations are presented for the calculation of lower and upper bounds on the optimal solution value of semi infinite programs sips involving smooth potentially nonconvex objective function and constraints the lower bounding problem is obtained by a formulation that combines the first and second order kkt necessary conditions of the lower level problem with a discretization of the parameter set the resulting mathematical program with equilibrium constraints mpec is a relaxation of the original sip and furnishes valid lower bounds if the parameter set is subdivided the optimal solution value of the lower bounding problem converges to the optimal solution value of the sip the upper bounding problem is based on convex and linear relaxations of the lower level problem resulting in a restriction of the sip if the parameter set is subdivided the constructed relaxations converge to the original lower level program the existence of sip slater points ensures convergence of the upper bounding problems to the optimal solution value of the sip several alternatives for the upper bounding problem are presented and discussed numerical results are presented for a number of test problems from the literature