wait and judge scenario optimization we consider convex optimization problems with uncertain probabilistically described constraints in this context scenario optimization is a well recognized methodology where a sample of the constraints is used to describe uncertainty one says that the scenario solution generalizes well or has a high robustness level if it satisfies most of the other constraints besides those in the sample over the past 10 years the main theoretical investigations on the scenario approach have related the robustness level of the scenario solution to the number of optimization variables this paper breaks into the new paradigm that the robustness level is a posteriori evaluated after the solution is computed and the actual number of the so called support constraints is assessed wait and judge a new theory is presented which shows that a posteriori observing k support constraints in dimension d k allows one to draw conclusions close to those obtainable when the problem is from the outset in dimension k this new theory provides evaluations of the robustness that largely outperform those carried out based on the number of optimization variables

doubly nonlinear thin film equations in one space dimension we consider a free boundary problem for a class of fourth order nonlinear parabolic equations which are degenerate both with respect to the unknown and to its third derivative the problem is relevant in the description of the surface tension driven spreading of a non newtonian liquid over a solid surface in the complete wetting regime relying solely on global and local energy estimates and on bernis inequalities we prove existence of solutions to this problem and obtain sharp upper bounds for the propagation of their support a necessary condition for the occurrence of waiting time phenomena is also derived

global existence and l estimates of solution for doubly nonlinear parabolic equation the existence uniqueness and l estimate of the weak solution to the initial boundary value problem for the doubly nonlinear parabolic equationut div u r u m 2 u au fx t uin 0 with zero boundary condition in a bounded domain rn are established in particular the behavior of the solution as t 0 and t is investigated

an implicit non linear time dependent equation has a solution has a solution u u w the operators s l and a t are maximal monotone from a real hilbert space v to its dual such that r 9 r are v coercive and a r are not degenerate a linear compact injection i embeds v to a real banach space w and each d r is the strongly monotone subdifferential of a continuous convex function i on w the function f is square integrable the functions w r v w are lipschitzian as v valued functions section 3 contains the theorems the main result is theorem 2 theorems 3 and 4 demonstrate the smoothing effect on the initial condition their proofs are given in section 4 they exploit the methods of di benedetto and showalter 4 who studied this equation in the case where d and a do not depend on time and v 0 but d can be degenerate grange and mignot studied a similar autonomous equation without v 7 saguez introduced the operator w 111 exploiting their methods by which he later proved together with bermudez and durany the solvability if 9j does not depend on time 131 in section 6 there is an example more examples in the time independent case as remarks and counterexamples on the uniqueness of the solution can be found in 4

global solution to a singular integro differential system related to the entropy balance this paper is devoted to the mathematical analysis of a thermodynamic model describing phase transitions with thermal memory in terms of an entropy equation and a momentum balance for the microforces the initial and boundary value problem is addressed for the related integro differential system of partial differential equations pdes existence and uniqueness continuous dependence on the data and regularity results are proved for the global solution in a finite time interval

on the sample size of random convex programs with structured dependence on the uncertainty the scenario approach provides an intuitive method to address chance constrained problems arising in control design for uncertain systems it addresses these problems by replacing the chance constraint with a finite number of sampled constraints scenarios the sample size critically depends on helly s dimension a quantity always upper bounded by the number of decision variables however this standard bound can lead to computationally expensive programs whose solutions are conservative in terms of cost and violation probability we derive improved bounds of helly s dimension for problems where the chance constraint has certain structural properties the improved bounds lower the number of scenarios required for these problems leading both to improved objective value and reduced computational complexity our results are generally applicable to randomized model predictive control of chance constrained linear systems with additive uncertainty and affine disturbance feedback

distributionally robust optimization under moment uncertainty with application to data driven problems stochastic programming can effectively describe many decision making problems in uncertain environments unfortunately such programs are often computationally demanding to solve in addition their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters in this paper we propose a model that describes uncertainty in both the distribution form discrete gaussian exponential etc and moments mean and covariance matrix we demonstrate that for a wide range of cost functions the associated distributionally robust or min max stochastic program can be solved efficiently furthermore by deriving a new confidence region for the mean and the covariance matrix of a random vector we provide probabilistic arguments for using our model in problems that rely heavily on historical data these arguments are confirmed in a practical example of portfolio selection where our framework leads to better performing policies on the true distribution underlying the daily returns of financial assets

existence for a doubly nonlinear volterra equation abstract we consider a doubly nonlinear volterra equation involving a nonsmooth kernel and two possibly degenerate monotone operators by exploiting an implicit time discretization procedure we obtain the existence of a global strong solution and extend to the nonlocal in time situation some former results by colli p colli on some doubly nonlinear evolution equations in banach spaces japan j indust appl math 9 2 1992 181 203

the scenario approach to robust control design this paper proposes a new probabilistic solution framework for robust control analysis and synthesis problems that can be expressed in the form of minimization of a linear objective subject to convex constraints parameterized by uncertainty terms this includes the wide class of np hard control problems representable by means of parameter dependent linear matrix inequalities lmis it is shown in this paper that by appropriate sampling of the constraints one obtains a standard convex optimization problem the scenario problem whose solution is approximately feasible for the original usually infinite set of constraints i e the measure of the set of original constraints that are violated by the scenario solution rapidly decreases to zero as the number of samples is increased we provide an explicit and efficient bound on the number of samples required to attain a priori specified levels of probabilistic guarantee of robustness a rich family of control problems which are in general hard to solve in a deterministically robust sense is therefore amenable to polynomial time solution if robustness is intended in the proposed risk adjusted sense

on spanners and lightweight spanners of geometric graphs we consider the problem of computing spanners of euclidean and unit disk graphs embedded in the two dimensional euclidean plane we are particularly interested in spanners that possess useful properties such as planarity bounded degree and or light weight such spanners have been extensively studied in the area of computational geometry and have been used as the building block for constructing efficient and reliable wireless network communication topologies we study the above problem under two computational models the centralized and the distributed model in the distributed model we focus on algorithms that are local such algorithms are suitable for the relevant applications e g wireless computing under the centralized model we present an o n lg n time algorithm that computes a bounded degree plane spanner of a complete euclidean graph where n is the number of points in the graph both upper bounds on the degree and the stretch factor significantly improve the previous bounds we extend this algorithm to compute a bounded degree plane lightweight spanner of a complete euclidean graph under the distributed model we give the first local algorithm for computing a spanner of a unit disk graph that is of bounded degree and plane the upper bounds on the degree stretch factor and the locality of the algorithm dramatically improve the previous results as shown in the paper this algorithm can also be extended to compute a bounded degree plane lightweight spanner of a unit disk graph our algorithms rely on structural and geometric results that we develop in this paper

a constraint sampling approach for multi stage robust optimization we propose a tractable approximation scheme for convex not necessarily linear multi stage robust optimization problems we approximate the adaptive decisions by finite linear combinations of prescribed basis functions and demonstrate how one can optimize over these decision rules at low computational cost through constraint randomization we obtain a priori probabilistic guarantees on the feasibility properties of the optimal decision rule by applying existing constraint sampling techniques to the semi infinite problem arising from the decision rule approximation we demonstrate that for a suitable choice of basis functions the approximation converges as the size of the basis and the number of sampled constraints tend to infinity the approach yields an algorithm parameterized in the basis size the probability of constraint violation and the confidence that this probability will not be exceeded these three parameters serve to tune the trade off between optimality and feasibility of the decision rules and the computational cost of the algorithm we assess the convergence and scalability properties of our approach in the context of two inventory management problems

discretization with variable time steps of an evolution equation with a positive type memory term we consider the linearized form of an evolution equation used to model viscoelasticity and heat conduction in materials with memory the equation is discretized in time using several schemes based on the backward euler and crank nicolson methods and incorporating appropriate quadratures for the memory term by permitting variable time steps we remove a restriction present in earlier related work by the first two authors

fast fast algorithm for the scenario technique the scenario approach is a recently introduced method to obtain feasible solutions to chance constrained optimization problems based on random sampling it has been noted that the sample complexity of the scenario approach rapidly increases with the number of optimization variables and this may pose a hurdle to its applicability to medium and large scale problems we here introduce the fast algorithm for the scenario technique a variant of the scenario optimization algorithm with reduced sample complexity subject classifications stochastic programming chance constrained optimization randomized algorithms sample based methods scenario approach area of review optimization

on a doubly nonlinear diffusion equation in hydrology on considere l equation aux derivees partielles non lineaires u t d u u x x ou d et satisfont certaines hypotheses on etudie le probleme de neumann sur 1 1 avec les conditions aux limites naturelles d u u x 1 t 0 pour t 0 le probleme de cauchy et un probleme de cauchy dirichlet sur 0 avec les conditions aux limites u 0 t a pour t 0 a 0 1

uncertain convex programs randomized solutions and confidence levels many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or instance parameter two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty robust optimization and chance constrained optimization robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances in chance constrained optimization a probability distribution is instead assumed on the uncertain parameters and the constraints are enforced up to a pre specified level of probability unfortunately however both approaches lead to computationally intractable problem formulations

semilinear subdiffusion with memory in the one dimensional case abstract for 0 1 we consider the semilinear integro differential equation on the one dimensional domain a b in the unknown u u x t d t u l 1 u 0 t k t s l 2 u s d s f x t u g x t where d t is the caputo fractional derivative and l 1 and l 2 are uniform elliptic operators with time dependent smooth coefficients under certain structural conditions on the nonlinearity f the global existence and uniqueness of classical solutions to the related initial boundary value problems are established via the so called continuation argument approach the key point is looking for suitable a priori estimates of the solution in the fractional holder spaces

blow up phenomena for polytropic equation with inhomogeneous density and source the subject of this investigation is the blow up phenomena of the positive solutions of the mixed problem for the one dimensional polytropic filtration equation with inhomogeneous density and source it is shown that under certain conditions on the nonlinearities and data blow up will occur at some finite time note that the technique applied for the proof does not use the zel dovich kompaneets barenblatt solutions since the construction of such type of function is more complicated in our case therefore we obtain a result by multiplying on a special factor which has convenient properties in particular by choosing the parameters of the factor and using the properties of the solution we obtain the inequality which allows us to show the blow up phenomena

doubly nonlinear equations with unbounded operators abstract the solvability of the evolution system v t b t u t f t v t a t u t 0 a t are bounded possibly degenerate subdifferentials and b t are unbounded subdifferentials

on safe tractable approximations of chance constraints abstract a natural way to handle optimization problem with data affected by stochastic uncertainty is to pass to a chance constrained version of the problem where candidate solutions should satisfy the randomly perturbed constraints with probability at least 1 while being attractive from modeling viewpoint chance constrained problems as they are are in general computationally intractable in this survey paper we overview several simulation based and simulation free computationally tractable approximations of chance constrained convex programs primarily those of chance constrained linear conic quadratic and semidefinite programming

solving the nonlinear richards equation model with adaptive domain decomposition modeling the transport processes in a vadose zone plays an important role in predicting the reactions of soil biotopes to anthropogenic activity e g modeling contaminant transport the effect of the soil water regime on changes in soil structure and composition etc water flow is governed by the richards equation while the constitutive laws are typically supplied by the van genuchten model which can be understood as a pore size distribution function certain materials with dominantly uniform pore sizes e g coarse grained materials can exhibit ranges of constitutive function values within several orders of magnitude possibly beyond the length of real numbers that computers can handle thus a numerical approximation of the richards equation often requires the solution of systems of equations that cannot be solved on computer arithmetics an appropriate domain decomposition into subdomains that cover only a limited range of constitutive function values and that will change adaptively reflecting the time progress of the model will enable an effective and reliable solution of this problem parts of this problem have already been treated in our work kuraz et al 2014 this paper focuses on improving the performance of a nonlinear solver by locating the areas with abrupt changes in the solution

on safe tractable approximations of chance constrained linear matrix inequalities in the paper we consider the chance constrained version of an affinely perturbed linear matrix inequality lmi constraint assuming the primitive perturbations to be independent with light tail distributions e g bounded or gaussian constraints of this type playing a central role in chance constrained linear conic quadratic semidefinite programming are typically computationally intractable the goal of this paper is to develop a tractable approximation to these chance constraints our approximation is based on measure concentration results and is given by an explicit system of lmis thus the approximation is computationally tractable moreover it is also safe meaning that a feasible solution of the approximation is feasible for the chance constraint

improved local algorithms for spanner construction let s be a set of n points in the plane let e be the complete euclidean graph whose point set is s and let g be the delaunay triangulation of s we present a very simple local algorithm that given g constructs a subgraph of g of degree at most 11 that is a geometric spanner of g with stretch factor 2 86 and hence a geometric spanner of e with stretch factor 7 this algorithm gives an o nlgn time centralized algorithm for constructing a subgraph of g that is a geometric spanner of e of degree at most 11 and stretch factor 7

the brezis ekeland principle for doubly nonlinear equations the celebrated brezis ekeland principle c r acad sci paris ser a b 282 1976 pp ai a1197 a1198 aii and a971 a974 characterizes trajectories of nonautonomous gradient flows of convex functionals as solutions to suitable minimization problems this note extends this characterization to doubly nonlinear evolution equations driven by convex potentials the characterization is exploited in order to establish approximation results for gradient flows doubly nonlinear equations and rate independent evolutions

the blow up phenomenon for degenerate parabolic equations with variable coefficients and nonlinear source abstract the cauchy problem for a degenerate parabolic equation with a source and variable coefficient of the form u t d i v x u m 1 d u 1 d u u p is studied global in time existence and nonexistence conditions are found for a solution to the cauchy problem exact estimates of a solution are obtained in the case of global solvability a sharp universal i e independent of the initial function estimate of a solution near the blow up time is obtained

the exact feasibility of randomized solutions of uncertain convex programs many optimization problems are naturally delivered in an uncertain framework and one would like to exercise prudence against the uncertainty elements present in the problem in previous contributions it has been shown that solutions to uncertain convex programs that bear a high probability to satisfy uncertain constraints can be obtained at low computational cost through constraint randomization in this paper we establish new feasibility results for randomized algorithms specifically the exact feasibility for the class of the so called fully supported problems is obtained it turns out that all fully supported problems share the same feasibility properties revealing a deep kinship among problems of this class it is further proven that the feasibility of the randomized solutions for all other convex programs can be bounded based on the feasibility for the prototype class of fully supported problems the feasibility result of this paper outperforms previous bounds and is not improvable because it is exact for fully supported problems

distributionally robust joint chance constraints with second order moment information we develop tractable semidefinite programming based approximations for distributionally robust individual and joint chance constraints assuming that only the first and second order moments as well as the support of the uncertain parameters are given it is known that robust chance constraints can be conservatively approximated by worst case conditional value at risk cvar constraints we first prove that this approximation is exact for robust individual chance constraints with concave or not necessarily concave quadratic constraint functions and we demonstrate that the worst case cvar can be computed efficiently for these classes of constraint functions next we study the worst case cvar approximation for joint chance constraints this approximation affords intuitive dual interpretations and is provably tighter than two popular benchmark approximations the tightness depends on a set of scaling parameters which can be tuned via a sequential convex optimization algorithm we show that the approximation becomes essentially exact when the scaling parameters are chosen optimally and that the worst case cvar can be evaluated efficiently if the scaling parameters are kept constant we evaluate our joint chance constraint approximation in the context of a dynamic water reservoir control problem and numerically demonstrate its superiority over the two benchmark approximations

existence results for doubly nonlinear higher order parabolic equations on unbounded domains on demontre un resultat d existence pour le probleme aux valeurs limites et initiales u q 1 sign u t a 1 u f sur 0 t q u satisfait des conditions de dirichlet sur 0 t u u 0 sur pour t 0 a 1 u 1 m m d d u p 1 sign d u pour p q 1 m n 1 et r n un ensemble ouvert

randomized strategies for probabilistic solutions of uncertain feasibility and optimization problems in this paper we study two general semi infinite programming problems by means of a randomized strategy based on statistical learning theory the sample size results obtained with this approach are generally considered to be very conservative by the control community the first main contribution of this paper is to demonstrate that this is not necessarily the case utilizing as a starting point one sided results from statistical learning theory we obtain bounds on the number of required samples that are manageable for ldquoreasonablerdquo values of probabilistic confidence and accuracy in particular we show that the number of required samples grows with the accuracy parameter epsiv as 1 epsivln 1 epsiv and this is a significant improvement when compared to the existing bounds which depend on 1 epsiv2ln 1 epsiv2 secondly we present new results for optimization and feasibility problems involving boolean expressions consisting of polynomials in this case when the accuracy parameter is sufficiently small an explicit bound that only depends on the number of decision variables and on the confidence and accuracy parameters is presented for convex optimization problems we also prove that the required sample size is inversely proportional to the accuracy for fixed confidence thirdly we propose a randomized algorithm that provides a probabilistic solution circumventing the potential conservatism of the bounds previously derived

regularity in sobolev spaces for doubly nonlinear parabolic equations abstract the doubly nonlinear parabolic equation u t div u m 1 u p 2 u m 1 u m 1 m p 1 1 is considered in several dimensions and regularity results in fractional order sobolev spaces are obtained the main tools in the proof are a difference quotient technique and the imbedding theorem of nikolskii spaces into sobolev spaces

scenario min max optimization and the risk of empirical costs we consider convex optimization problems in the presence of stochastic uncertainty the min max sample based solution is the solution obtained by minimizing the max of the cost functions corresponding to a finite sample of the uncertainty parameter the empirical costs are instead the cost values that the solution incurs for the various parameter realizations that have been sampled our goal is to evaluate the risks associated with the empirical costs where the risk associated with a cost is the probability that the cost is exceeded when a new realization of the uncertainty parameter is seen this task is accomplished without resorting to uncertainty realizations other than those used in optimization the theoretical result proved in this paper is that these risks form a random vector whose probability distribution is an ordered dirichlet distribution irrespective of the probability measure of the stochastic uncertainty parameter this result provides a distribution free characterization of the risks as

a coverage theory for least squares summary a sensible use of an estimation method requires that assessment criteria for the quality of the estimate be available we present a coverage theory for the least squares estimate by suitably modifying the empirical costs one constructs statistics that are guaranteed to cover with known probability the cost associated with a next still unseen member of the population all results of this paper are distribution free and can be applied to least squares problems in use across a variety of fields

an integer programming approach for linear programs with probabilistic constraints linear programs with joint probabilistic constraints pclp are difficult to solve because the feasible region is not convex we consider a special case of pclp in which only the right hand side is random and this random vector has a finite distribution we give a mixed integer programming formulation for this special case and study the relaxation corresponding to a single row of the probabilistic constraint we obtain two strengthened formulations as a byproduct of this analysis we obtain new results for the previously studied mixing set subject to an additional knapsack inequality we present computational results which indicate that by using our strengthened formulations instances that are considerably larger than have been considered before can be solved to optimality

nonlocal in time problem for anisotropic parabolic equations with variable exponents of nonlinearities abstract in this study we consider nonlinear parabolic equations with variable exponents of the nonlinearities the nonlocal problem for these equations is investigated and the existence and uniqueness theorems are proved for the problem

a provisioning problem with stochastic payments abstract we consider the problem of determining the minimal requirement one must establish in order to meet a series of future random payments it is shown in a very general setting that this problem can be recast as a chance constrained model and how the technique of sample average approximation can be employed to find solutions we also use comonotonic theory to analyze analytical approximations in a restricted gaussian setting our numerical illustrations demonstrate that the sample average approximation is a viable and efficient way to solve the stated problem generally and outperforms the analytical approximations in passing we present a result that is related to stein s famous lemma stein 1981 and is of interest in itself

time discretization and global solution for a doubly nonlinear volterra equation this note addresses the analysis of an abstract doubly nonlinear volterra equation with a nonsmooth kernel and possibly unbounded and degenerate operators by exploiting a suitable implicit time discretization technique we obtain the existence of a global strong solution as a by product the discrete scheme is proved to be conditionally stable and convergent

on the evolution p laplacian with nonlocal memory abstract we study the homogeneous dirichlet problem for the evolution p laplacian with the nonlocal memory term 0 1 u t p u 0 t g t s p u x s d s x t u f x t in q 0 t where r n is a bounded domain g and f are given functions it is proved that for max 1 2 n n 2 p g g l 2 0 t and u 0 w 0 1 p f l 2 q the problem admits a weak solution which is global or local in time in dependence on the growth rate of x t s as s conditions of uniqueness are established it is proved that for p 2 and s x t s 0 the disturbances from the data propagate with finite speed and the waiting time effect is possible we present simple explicit solutions that show the failure of the maximum and comparison principles for the solutions of equation 0 1

convex approximations of chance constrained programs we consider a chance constrained problem where one seeks to minimize a convex objective over solutions satisfying with a given close to one probability a system of randomly perturbed convex constraints this problem may happen to be computationally intractable our goal is to build its computationally tractable approximation i e an efficiently solvable deterministic optimization program with the feasible set contained in the chance constrained problem we construct a general class of such convex conservative approximations of the corresponding chance constrained problem moreover under the assumptions that the constraints are affine in the perturbations and the entries in the perturbation vector are independent of each other random variables we build a large deviation type approximation referred to as bernstein approximation of the chance constrained problem this approximation is convex and efficiently solvable we propose a simulation based scheme for bounding the optimal value in the chance constrained problem and report numerical experiments aimed at comparing the bernstein and well known scenario approximation approaches finally we extend our construction to the case of ambiguous chance constrained problems where the random perturbations are independent with the collection of distributions known to belong to a given convex compact set rather than to be known exactly while the chance constraint should be satisfied for every distribution given by this set

a sample approximation approach for optimization with probabilistic constraints we study approximations of optimization problems with probabilistic constraints in which the original distribution of the underlying random vector is replaced with an empirical distribution obtained from a random sample we show that such a sample approximation problem with a risk level larger than the required risk level will yield a lower bound to the true optimal value with probability approaching one exponentially fast this leads to an a priori estimate of the sample size required to have high confidence that the sample approximation will yield a lower bound we then provide conditions under which solving a sample approximation problem with a risk level smaller than the required risk level will yield feasible solutions to the original problem with high probability once again we obtain a priori estimates on the sample size required to obtain high confidence that the sample approximation problem will yield a feasible solution to the original problem finally we present numerical illustrations of how these results can be used to obtain feasible solutions and optimality bounds for optimization problems with probabilistic constraints

an abstract doubly nonlinear equation with a measure as initial value abstract the solvability of the abstract implicit nonlinear nonautonomous differential equation a t u t b t u t c t u t f t will be investigated in the case of a measure as an initial value it will be shown that this problem has a solution if the inner product of a t x and b t x c t x is bounded below

structure of boundary blow up for higher order quasilinear parabolic equations singularity formation phenomena for 2 m th order quasilinear parabolic equations are studied by using energy estimates related to saint venant s principle sharp estimates of propagation of singularities generated by boundary global and regional blow up regimes are established

interval predictor models identification and reliability this paper addresses the problem of constructing reliable interval predictors directly from observed data differently from standard predictor models interval predictors return a prediction interval as opposed to a single prediction value we show that in a stationary and independent observations framework the reliability of the model that is the probability that the future system output falls in the predicted interval is guaranteed a priori by an explicit and non asymptotic formula with no further assumptions on the structure of the unknown mechanism that generates the data this fact stems from a key result derived in this paper which relates at a fundamental level the reliability of the model to its complexity and to the amount of available information number of observed data

a class of nonlinear elliptic parabolic equations with time dependent constraints on etudie les equations d evolutions non lineaires de la forme f t du t dt t v t v t bu t 0 t t dans un espace de hilbert h ou t est la sous differentielle d une fonction convexe t definie sur h b est un operateur monotone maximal dans h on considere l existence et l unicite des solutions ainsi que des estimations a priori

finite speed of propagation and asymptotic rates for some nonlinear higher order parabolic equations with absorption the energy solutions to the equation have finite speed of propagation if l or l if 1 vq t is uniformly bounded for t 0 localisation property and if q sharp upper bounds of the interface or free boundary are obtained we use a weighted energy method the weights being powers of the distance to a variable half space we also study decay rates as t and extinction in finite time for bounded and unbounded domains with null dirichlet boundary conditions our equation includes the porous media equation with absorption analogous results hold if m is replaced by an appropriate quasilinear elliptic operator

fast greedy algorithms for constructing sparse geometric spanners given a set v of n points in ir d and a real constant t 1 we present the first o nlog n time algorithm to compute a geometric t spanner on v a geometric t spanner on v is a connected graph g v e with edge weights equal to the euclidean distances between the endpoints and with the property that for all u v in v the distance between u and v in g is at most t times the euclidean distance between u and v the spanner output by the algorithm has o n edges and weight o 1 cdot wt mst and its degree is bounded by a constant

doubly nonlinear periodic problems with unbounded operators abstract the solvability of the evolution system v t b t u t f t v t a t u t 0 t t with the periodic condition v 0 v t is investigated in the case where a t are bounded possibly degenerate subdifferentials and b t are unbounded subdifferentials

risk and complexity in scenario optimization scenario optimization is a broad methodology to perform optimization based on empirical knowledge one collects previous cases called scenarios for the set up in which optimization is being performed and makes a decision that is optimal for the cases that have been collected for convex optimization a solid theory has been developed that provides guarantees of performance and constraint satisfaction of the scenario solution in this paper we open a new direction of investigation the risk that a performance is not achieved or that constraints are violated is studied jointly with the complexity as precisely defined in the paper of the solution it is shown that the joint probability distribution of risk and complexity is concentrated in such a way that the complexity carries fundamental information to tightly judge the risk this result is obtained without requiring extra knowledge on the underlying optimization problem than that carried by the scenarios in particular no extra knowledge on the distribution by which scenarios are generated is assumed so that the result is broadly applicable this deep seated result unveils a fundamental and general structure of data driven optimization and suggests practical approaches for risk assessment

instantaneous shrinking of the support of energy solutions the cauchy problem for a class of nonlinear diffusion reaction equations is studied the equations may be classified as being of degenerate parabolic type it is shown that under certain conditions solutions of the problem exhibit instantaneous shrinking this is to say at any positive time the spatial support of the solution is bounded above although the support of the initial data function is not we also provide some estimates of the behavior of the free boundary

on the connection between compression learning and scenario based single stage and cascading optimization problems we investigate the connections between compression learning and scenario based optimization we first show how to strengthen or relax the consistency assumption at the basis of compression learning and provide novel learnability conditions for the underlying algorithms we then consider different constrained optimization problems affected by uncertainty represented by means of scenarios we show that the compression learning perspective provides a unifying framework for scenario based optimization since the issue of providing guarantees on the probability of constraint violation reduces to a learning problem for an appropriately chosen algorithm that satisfies some consistency assumption to illustrate this we revisit the scenario approach within the developed context moreover using the compression learning machinery we provide novel results on the probability of constraint violation for the class of cascading optimization problems

periodic solutions for doubly nonlinear evolution equations abstract we discuss the existence of periodic solution for the doubly nonlinear evolution equation a u t u t f t governed by a maximal monotone operator a and a subdifferential operator in a hilbert space h as the corresponding cauchy problem cannot be expected to be uniquely solvable the standard approach based on the poincare map may genuinely fail in order to overcome this difficulty we firstly address some approximate problems relying on a specific approximate periodicity condition then periodic solutions for the original problem are obtained by establishing energy estimates and by performing a limiting procedure as a by product a structural stability analysis is presented for the periodic problem and an application to nonlinear pdes is provided

hyperbolic phenomena in a strongly degenerate parabolic equation we consider the equation u t u u x x where 0 and where is a strictly increasing function with lim s we solve the associated cauchy problem for an increasing initial function and discuss to what extent the solution behaves qualitatively like solutions of the first order conservation law u t u x equations of this type arise for example in the theory of phase transitions where the corresponding free energy functional has a linear growth rate with respect to the gradient

distribution free prediction sets this article introduces a new approach to prediction by bringing together two different nonparametric ideas distribution free inference and nonparametric smoothing specifically we consider the problem of constructing nonparametric tolerance prediction sets we start from the general conformal prediction approach and we use a kernel density estimator as a measure of agreement between a sample point and the underlying distribution the resulting prediction set is shown to be closely related to plug in density level sets with carefully chosen cutoff values under standard smoothness conditions we get an asymptotic efficiency result that is near optimal for a wide range of function classes but the coverage is guaranteed whether or not the smoothness conditions hold and regardless of the sample size the performance of our method is investigated through simulation studies and illustrated in a real data example

randomized solutions to convex programs with multiple chance constraints the scenario based optimization approach scenario approach provides an intuitive way of approximating the solution to chance constrained optimization programs based on finding the optimal solution under a finite number of sampled outcomes of the uncertainty scenarios a key merit of this approach is that it neither requires explicit knowledge of the uncertainty set as in robust optimization nor of its probability distribution as in stochastic optimization the scenario approach is also computationally efficient because it only requires the solution to a convex optimization program even if the original chance constrained problem is nonconvex recent research has obtained a rigorous foundation for the scenario approach by establishing a direct link between the number of scenarios and bounds on the constraint violation probability these bounds are tight in the general case of an uncertain optimization problem with a single chance constraint this paper shows that the bounds can be improved

a scenario approach for non convex control design randomized optimization is an established tool for control design with modulated robustness while for uncertain convex programs there exist efficient randomized approaches this is not the case for non convex problems methods based on statistical learning theory are applicable to non convex problems but they usually are conservative in achieving the desired probabilistic guarantees in this paper we derive a novel scenario approach for a wide class of random non convex programs with a sample complexity similar to that of uncertain convex programs and with probabilistic guarantees that hold not only for the optimal solution of the scenario program but for all feasible solutions inside a set of a priori chosen complexity we also address measure theoretic issues for uncertain convex and non convex programs among the family of non convex control design problems that can be addressed via randomization we apply our scenario approach to stochastic model predictive control for chance constrained nonlinear control affine systems

uniqueness of renormalized solutions of degenerate elliptic parabolic problems we consider a general class of degenerate elliptic paraboic problems associated with the equation b v t div a v dv f using kruzhkov s method of doubling variables both in space and time we prove uniqueness and a comparison principle in l 1 for renormalized solutions

randomized methods for design of uncertain systems in this paper we study randomized methods for feedback design of uncertain systems the first contribution is to derive the sample complexity of various constrained control problems in particular we show the key role played by the binomial distribution and related tail inequalities and compute the sample complexity this contribution significantly improves the existing results by reducing the number of required samples in the randomized algorithm these results are then applied to the analysis of worst case performance and design with robust optimization the second contribution of the paper is to introduce a general class of sequential algorithms denoted as sequential probabilistic validation spv in these sequential algorithms at each iteration a candidate solution is probabilistically validated and corrected if necessary to meet the required specifications the results we derive provide the sample complexity which guarantees that the solutions obtained with spv algorithms meet some pre specified probabilistic accuracy and confidence the performance of these algorithms is illustrated and compared with other existing methods using a numerical example dealing with robust system identification

existence and convergence theorems for doubly nonlinear partial differential equations of elliptic parabolic type here t 0 v u ax au ax au ax and b is a map from rn x 0 t x r x rn into r for each i 1 n satisfying certain conditions to be specified at a later time a precise formulation of 1 1 is given in section 3 for given u0 and f we study the existence of weak solutions of 1 1 although doubly nonlinear equations of the type 1 1 are of mathematical interest in their own right they arise as a model for a variety of diffusion problems in particular they comprise in a unifying scheme free boundary problems of different natures an example situation is nonstationary saturated unsaturated flow of an incompressible fluid through a porous medium in the case of time dependent water levels a weak formulation of this problem leads to equations of the type 1 1 see 2 however a large held of application for our results is the diffusion problem involving a solid liquid phase change of the stefan type an architypical example is heat transfer during solidification in a nonhomogeneous 205

the smoothing property for a class of doubly nonlinear parabolic equations we consider a class of doubly nonlinear parabolic equations used in modeling free boundaries with a finite speed of propagation we prove that nonnegative weak solutions satisfy a smoothing property this is a well known feature in some particular cases such as the porous medium equation or the parabolic p laplace equation the result is obtained via regularization and a comparison theorem

nonsmooth data error estimates for approximations of an evolution equation with a positive type memory term we study the numerical approximation of an integro differential equation which is intermediate between the heat and wave equations the proposed discretization uses convolution quadrature based on the first and second order backward difference methods in time and piecewise linear finite elements in space optimal order error bounds in terms of the initial data and the inhomogeneity are shown for positive times without assumptions of spatial regularity of the data

a pseudo monotonicity adapted to doubly nonlinear elliptic parabolic equations the case where b is an unbounded linear operator was considered 1rst by bardos and br7 ezis 4 in the nonlinear case raviart 20 grange and mignot 13 dibenedetto and showalter 11 proved existence results assuming that a and b are at least monotone operators and b is compact alt and luckhaus 1 investigated the case of a non compact operator b assuming a is strongly monotone their work was extended more recently by ka cur 14 filo and ka cur 12 or zadrzy7 nska and zajb aczkowski 26 berm7 udez et al 7 devoted their work to the case where b is compact and strongly monotone and a is pseudo monotone we are interested in the same case excepted that b is no longer assumed to be strongly monotone thus the equation may

flow of oil and water in a porous medium abstract the flow of two immiscible and incompressible fluids in a porous medium is described by a system of quasilinear degenerate partial differential equations in this paper the existence of a weak solution by regularization is shown

p euler equations and p navier stokes equations abstract we propose in this work new systems of equations which we call p euler equations and p navier stokes equations p euler equations are derived as the euler lagrange equations for the action represented by the benamou brenier characterization of wasserstein p distances with incompressibility constraint p euler equations have similar structures with the usual euler equations but the momentum is the signed p 1 th power of the velocity in the 2d case the p euler equations have streamfunction vorticity formulation where the vorticity is given by the p laplacian of the streamfunction by adding diffusion presented by laplacian of the velocity we obtain what we call p navier stokes equations if p the a priori energy estimates for the velocity and momentum have dual symmetries using these energy estimates and a time shift estimate we show the global existence of weak solutions for the p navier stokes equations in r d for p and p d 2 through a compactness criterion

on the road between robust optimization and the scenario approach for chance constrained optimization problems we propose a new method for solving chance constrained optimization problems that lies between robust optimization and scenario based methods our method does not require prior knowledge of the underlying probability distribution as in robust optimization methods nor is it based entirely on randomization as in the scenario approach it instead involves solving a robust optimization problem with bounded uncertainty where the uncertainty bounds are randomized and are computed using the scenario approach to guarantee that the resulting robust problem is solvable we impose certain assumptions on the dependency of the constraint functions with respect to the uncertainty and show that tractability is ensured for a wide class of systems our results lead immediately to guidelines under which the proposed methodology or the scenario approach is preferable in terms of providing less conservative guarantees or reducing the computational cost

on uniform and decay estimates for unbounded solutions of partial differential equations abstract we prove that if a function u satisfies certain integral estimates of energy type then it verifies estimates of the type u t l r c u 0 l r 0 0 t 1 t 0 where r and r 0 are exponents that appear in these integral estimates and 1 and 0 are positive constants that can be expressed in terms of these constants we will see that in some cases r r 0 supercontractive estimates and hence a regularizing effect on u appears while in other cases this improvement of regularity does not appear since r r 0 in any case we prove that the l r norm of u decays in time for t large since it verifies the previous estimate we show how to apply this result to obtain new estimates of this kind for the solutions of many nonlinear parabolic equations this new method allows also to explain the reason of the similar behavior of the solutions of very different parabolic problems finally we study sufficient conditions for extinction in finite time