existence uniqueness and l q estimates for the stokes problem in an exterior domain as is well known the study of the properties of solutions to the stokes system plays an essential role in the mathematical theory of viscous fluid flows governed by the navier stokes equations it is not art overstatement to say that existence uniqueness and a priori estimates in sobolev spaces for the stokes system are fundamental in every basic question related to the navier stokes equations such as existence regularity asymptotic behavior in time etc see e g ladyzhenskaya 1969 solonni ov 1977 and te am 1977 so far however the stokes system has been fully analyzed only for bounded domains see sobolevskn 1960 ladvznenskaya 1959 solonnu ov 1960 vorovlch yot oowcr 1961 cattabrma 1961 in particular in the paper of cattabriga a complete treatment is given in the sobotev spaces w l 1 1 q c for space dimension n 3 cattabriga s results essentially state that if 2 is a bounded domain in r a of class c m m max l 2 2 l 1 then given f e wt q y2 3 gc wt l q o q e wl 2 1 q q 8o a 1 q there exists one and only one distributional solution v c w1 z q q 3 p e wt l q o to the stokes system

the c 3 theorem and a d 2 algorithm for large scale stochastic mixed integer programming set convexification this paper considers the two stage stochastic integer programming problem with an emphasis on instances in which integer variables appear in the second stage drawing heavily on the theory of disjunctive programming we characterize convexifications of the second stage problem and develop a decomposition based algorithm for the solution of such problems in particular we verify that problems with fixed recourse are characterized by scenario dependent second stage convexifications that have a great deal in common we refer to this characterization as the c3 common cut coefficients theorem based on the c3 theorem we develop a decomposition algorithm which we refer to as disjunctive decomposition d2 in this new class of algorithms we work with master and subproblems that result from convexifications of two coupled disjunctive programs we show that when the second stage consists of 0 1 milp problems we can obtain accurate second stage objective function estimates after finitely many steps this result implies the convergence of the d2 algorithm

semidirect products and reduction in mechanics this paper shows how to reduce a hamiltonian system on the cotangent bundle of a lie group to a hamiltonian system in the dual of the lie algebra of a semidirect product the procedure simplifies unifies and extends work of greene guillemin holm holmes kupershmidt marsden morrison ratiu sternberg and others the heavy top compressible fluids magnetohydrodynamics elasticity the maxwell vlasov equations and multifluid plasmas are presented as examples starting with lagrangian variables our method explains in a direct way why semidirect products occur so frequently in examples it also provides a framework for the systematic introduction of clebsch or canonical variables

a finite branch and bound algorithm for two stage stochastic integer programs this paper addresses a general class of two stage stochastic programs with integer recourse and discrete distributions we exploit the structure of the value function of the second stage integer problem to develop a novel global optimization algorithm the proposed scheme departs from those in the current literature in that it avoids explicit enumeration of the search space while guaranteeing finite termination computational experiments on standard test problems indicate superior performance of the proposed algorithm in comparison to those in the existing literature

disjunctive decomposition for two stage stochastic mixed binary programs with random recourse this paper introduces disjunctive decomposition for two stage mixed 0 1 stochastic integer programs sips with random recourse disjunctive decomposition allows for cutting planes based on disjunctive programming to be generated for each scenario subproblem under a temporal decomposition setting of the sip problem a new class of valid inequalities for mixed 0 1 sip with random recourse is presented in particular we derive valid inequalities that allow for scenario subproblems for sip with random recourse but deterministic technology matrix and right hand side vector to share cut coefficients the valid inequalities are used to derive a disjunctive decomposition method whose derivation has been motivated by real life stochastic server location problems with random recourse which find many applications in operations research computational results with large scale instances to demonstrate the potential of the method are reported

traces for homogeneous sobolev spaces in infinite strip like domains abstract in this paper we construct a trace operator for homogeneous sobolev spaces defined on infinite strip like domains we identify an intrinsic seminorm on the resulting trace space that makes the trace operator bounded and allows us to construct a bounded right inverse the intrinsic seminorm involves two features not encountered in the trace theory of bounded lipschitz domains or half spaces first due to the strip like structure of the domain the boundary splits into two infinite disconnected components the traces onto each component are not completely independent and the intrinsic seminorm contains a term that measures the difference between the two traces second as in the usual trace theory there is a term in the seminorm measuring the fractional sobolev regularity of the trace functions with a difference quotient integral however the finite width of the strip like domain gives rise to a screening effect that bounds the range of the difference quotient perturbation the screened homogeneous fractional sobolev spaces defined by this screened seminorm on arbitrary open sets are of independent interest and we study their basic properties we conclude the paper with applications of the trace theory to partial differential equations

the dirichlet problem for nonlocal operators in this note we set up the elliptic and the parabolic dirichlet problem for linear nonlocal operators as opposed to the classical case of second order differential operators here the boundary data are prescribed on the complement of a given bounded set we formulate the problem in the classical framework of hilbert spaces and prove unique solvability using standard techniques like the fredholm alternative

on the convergence of cutting plane algorithms for a class of nonconvex mathematical programs we discuss the convergence of cutting plane algorithms for a class of nonconvex programs called the generalized lattice point problems glpp a set of sufficient conditions which guarantee finite convergence are presented although these conditions are usually difficult to enforce in a practical implementation they do illustrate the various factors that must be involved in a convergent rudimentary cutting plane algorithm a striking example of nonconvergence in which no subsequence converges to a feasible solution even when seemingly strong cutting planes are used is presented to show the effect of neglecting one such factor we give an application of our analysis to problems with multiple choice constraints and finally discuss a modification of cutting plane algorithms so as to make finite convergence more readily implementable

a polyhedral study of production ramping we give strong formulations of ramping constraints used to model the maximum change in production level for a generator or machine from one time period to the next and production limits for the two period case we give a complete description of the convex hull of the feasible solutions the two period inequalities can be readily used to strengthen ramping formulations without the need for separation for the general case we define exponential classes of multi period variable upper bound and multi period ramping inequalities and give conditions under which these inequalities define facets of ramping polyhedra finally we present exact polynomial separation algorithms for the inequalities and report computational experiments on using them in a branch and cut algorithm to solve unit commitment problems in power generation

a paradifferential approach for well posedness of the muskat problem we study the muskat problem for one fluid or two fluids with or without viscosity jump with or without rigid boundaries and in arbitrary space dimension d of the interface the muskat problem is scaling invariant in the sobolev space h s c mathbb r d where s c 1 frac d 2 employing a paradifferential approach we prove local well posedness for large data in any subcritical sobolev spaces h s mathbb r d s s c moreover the rigid boundaries are only required to be lipschitz and can have arbitrarily large variation the rayleigh taylor stability condition is assumed for the case of two fluids with viscosity jump but is proved to be automatically satisfied for the case of one fluid the starting point of this work is a reformulation solely in terms of the drichlet neumann operator the key elements of proofs are new paralinearization and contraction results for the drichlet neumann operator in rough domains

cutting planes for multistage stochastic integer programs this paper addresses the problem of finding cutting planes for multistage stochastic integer programs we give a general method for generating cutting planes for multistage stochastic integer programs based on combining inequalities that are valid for the individual scenarios we apply the method to generate cuts for a stochastic version of a dynamic knapsack problem and for stochastic lot sizing problems we give computational results which show that these new inequalities are very effective in a branch and cut algorithm

a modified lift and project procedure in recent years the lift and project approach has been used successfully within a branch and cut framework to solve large difficult pure and mixed 0 1 programs that have resisted solution efforts by pure branch and bound codes the approach uses a linear description in a higher dimensional space of the convex hull of the disjunctive set created by imposing one or several 0 1 conditions by solving a linear program derived from this higher dimensional representation the cut generating linear program cglp the standard lift and project procedure obtains a deepest cut in a well defined sense we propose a modification of cglp that allows us to generate not just one deepest cut but a class of cuts with desirable properties each at the cost of one extra pivot in the optimal tableau of the modified cglp

decomposition algorithms with parametric gomory cuts for two stage stochastic integer programs we consider a class of two stage stochastic integer programs with binary variables in the rst stage and general integer variables in the second stage we develop decomposition algorithms akin to the l shaped or benders methods by utilizing gomory cuts to obtain iteratively tighter approximations of the second stage integer programs we show that the proposed methodology is exible in that it allows several modes of implementation all of which lead to nitely convergent algorithms we illustrate our algorithms using examples from the literature we report computational results using the stochastic server location problem instances which suggest that our decomposition based approach scales better with increases in the number of scenarios than a state of the art solver which was used to solve the deterministic equivalent formulation

solving nonlinear single unit commitment problems with ramping constraints we present a dynamic programming algorithm for solving the single unit commitment 1uc problem with ramping constraints and arbitrary convex cost functions the algorithm is based on a new approach for efficiently solving the single unit economic dispatch ed problem with ramping constraints and arbitrary convex cost functions improving on previously known ones that were limited to piecewise linear functions for simple convex functions such as the quadratic ones typically used in applications the solution cost of all the involved ed problems consisting of finding an optimal

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

stochastic power generation unit commitment in electricity markets a novel formulation and a comparison of solution methods we propose a stochastic unit commitment model for a power generation company that takes part in an electricity spot market the relevant feature of this model is its detailed representation of the spot market during a whole week including seven day ahead market sessions and the corresponding adjustment market sessions the adjustment market sessions can be seen as an hour ahead market mechanism this representation takes into account the influence that the company s decisions exert on the market clearing price by means of a residual demand curve for each market session we introduce uncertainty in the form of several possible spot market outcomes for each day which leads to a weekly scenario tree the model also represents in detail the operation of the company s generation units the model leads to large scale mixed linear integer problems that are hard to solve with commercial optimizers this suggests the use of alternative solution methods we test four solution approaches with a realistic numerical example in the context of the spanish electricity spot market the first is a direct solution with a commercial optimizer which illustrates the mentioned limitations the second is a standard lagrangean relaxation algorithm the third and fourth methods are two original variants of benders decomposition for multistage stochastic integer programs the first benders decomposition algorithm builds approximations for the recourse function relaxing the integrality constraints of the subproblems the second variant strengthens these cuts by performing one iteration of the lagrangean of each subproblem we analyze the advantages of these four methods and compare the results

relative equilibria and bifurcations in the generalized van der waals 4d oscillator abstract a uniparametric 4 dof family of perturbed hamiltonian oscillators in 1 1 1 1 resonance is studied as a generalization for several models for perturbed keplerian systems normalization by lie transforms only first order is considered here as well as geometric reduction related to the invariants associated to the symmetries is used based on the previous work of the authors a description is given for the lower dimensional relative equilibria in such normalized systems for which we introduce in this context the new concept of moment polytope in addition bifurcations of relative equilibria corresponding to 3d tori are studied in some particular cases where we focus on hamiltonian hopf bifurcations and bifurcations in the 3d van der waals and zeeman systems

a polyhedral study of the semi continuous knapsack problem we study the convex hull of the feasible set of the semi continuous knapsack problem in which the variables belong to the union of two intervals such structure arises in a wide variety of important problems e g blending and portfolio selection we show how strong inequalities valid for the semi continuous knapsack polyhedron can be derived and used in a branch and cut scheme for problems with semi continuous variables to demonstrate the effectiveness of these inequalities which we call collectively semi continuous cuts we present computational results on real instances of the unit commitment problem as well as on a number of randomly generated instances of linear programming with semi continuous variables

decomposition with branch and cut approaches for two stage stochastic mixed integer programming decomposition has proved to be one of the more effective tools for the solution of large scale problems especially those arising in stochastic programming a decomposition method with wide applicability is benders decomposition which has been applied to both stochastic programming as well as integer programming problems however this method of decomposition relies on convexity of the value function of linear programming subproblems this paper is devoted to a class of problems in which the second stage subproblem s may impose integer restrictions on some variables the value function of such integer subproblem s is not convex and new approaches must be designed in this paper we discuss alternative decomposition methods in which the second stage integer subproblems are solved using branch and cut methods one of the main advantages of our decomposition scheme is that stochastic mixed integer programming smip problems can be solved by dividing a large problem into smaller mip subproblems that can be solved in parallel this paper lays the foundation for such decomposition methods for two stage stochastic mixed integer programs

the value of multistage stochastic programming in capacity planning under uncertainty this paper addresses a general class of capacity planning problems under uncertainty which arises for example in semiconductor tool purchase planning using a scenario tree to model the evolution of the uncertainties we develop a multistage stochastic integer programming formulation for the problem in contrast to earlier two stage approaches the multistage model allows for revision of the capacity expansion plan as more information regarding the uncertainties is revealed we provide analytical bounds for the value of multistage stochastic programming vms afforded over the two stage approach by exploiting a special substructure inherent in the problem we develop an efficient approximation scheme for the difficult multistage stochastic integer program and prove that the proposed scheme is asymptotically optimal computational experiments with realistic scale problem instances suggest that the vms for this class of problems is quite high moreover the quality and performance of the approximation scheme is very satisfactory fortunately this is more so for instances for which the vms is high

nondifferential reverse convex programs and facetial convexity cuts via a disjuntive characterization disjunctive programs can often be transcribed as reverse convex constrained problems with nondifferentiable constraints and unbounded feasible regions we consider this general class of nonconvex programs called reverse convex programs rcp and show that under quite general conditions the closure of the convex hull of the feasible region is polyhedral this development is then pursued from a more constructive standpoint in that for certain special reverse convex sets we specify a finite linear disjunction whose closed convex hull coincides with that of the special reverse convex set when interpreted in the context of convexity intersection cuts this provides the capability of generating any negative edge extension facet cut although this characterization is more clarifying than computationally oriented our development shows that if certain bounds are available then convexity intersection cuts can be strengthened relatively inexpensively

dynamics of the kirchhoff equations i coincident centers of gravity and buoyancy abstract we study the kirchhoff equations for a rigid body immersed in an incompressible irrotational inviscid fluid in the case that the centers of buoyancy and gravity coincide the resulting dynamical equations form a non canonical hamiltonian system with a six dimensional phase space which may be reduced to a four dimensional two degree of freedom canonical system using the two casimir invariants of motion restricting ourselves to ellipsoidal bodies we identify several completely integrable subcases in the general case we analyze existence linear and nonlinear stability and bifurcations of equilibria corresponding to steady translations and rotations including mixed modes involving simultaneous motion along two body axes some of which we show can be stable by perturbing from the axisymmetric integrable case we show that slightly asymmetric ellipsoids are typically non integrable and we investigate their dynamics with a view to using motions along homo and heteroclinic orbits to execute specific maneuvers in autonomous underwater vehicles

traces of weighted sobolev spaces old and new abstract we give short simple proofs of uspenski s results characterizing besov spaces as trace spaces of weighted sobolev spaces we generalize uspenski s results and prove the optimality of these generalizations we next show how classical results on the functional calculus in the besov spaces can be obtained as straightforward consequences of the theory of weighted sobolev spaces

mixed 0 1 programming by lift and project in a branch and cut framework we investigate the computational issues that need to be addressed when incorporating general cutting planes for mixed 0 1 programs into a branch and cut framework the cuts we use are of the lift and project variety some of the issues addressed have a theoretical answer but others are of an experimental nature and are settled by comparing alternatives on a set of test problems the resulting code is a robust solver for mixed 0 1 programs we compare it with several existing codes on a wide range of test problems it performs as well as or better than some of the best currently available mixed integer programming codes

a geometric buchberger algorithm for integer programming let ip a c denote the family of integer programs of the form min cx ax b x nn obtained by varying the right hand side vector b but keeping a and c fixed a test set for ipa c is a set of vectors in zn such that for each nonoptimal solution to a program in this family there is at least one element g in this set such that g has an improved cost value as compared to we describe a unique minimal test set for this family called the reduced grobner basis of ip a c an algorithm for its construction is presented which we call a geometric buchberger algorithm for integer programming and we show how an integer program may be solved using this test set the reduced grobner basis is then compared with some other known test sets from the literature we also indicate an easy procedure to construct test sets with respect to all cost functions for a matrix a zn 2 n of full row rank

on formulae decoupling the total variation of bv functions abstract in this paper we prove several formulae that enable one to capture the singular portion of the measure derivative of a function of bounded variation as a limit of non local functionals one special case shows that rescalings of the fractional laplacian of a function u s b v converge strictly to the singular portion of d u

mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions in this paper we study the linear bond based nonlocal peridynamic models with a particular focus on problems associated with nonstandard nonlocal displacement loading conditions both stationary and time dependent problems are considered for a one dimensional scalar equation defined on a finite bar and for a two dimensional system defined on a square the related peridynamic operators and associated functional spaces are analyzed applications to the numerical analysis of the finite dimensional approximations to peridynamic models are also illustrated

nonlinear stability of the equilibria in a double bar rotating system we study the nonlinear stability of the equilibria corresponding to the motion of a particle orbiting around a two finite orthogonal straight segment the potential is a logarithmic function and may be considered as an approximation to the one generated by irregular celestial bodies using arnold s theorem for non definite quadratic forms we determine the nonlinear stability of the equilibria for all values of the parameter of the problem moreover the resonant cases are determined and the stability is investigated

solving stochastic programs with integer recourse by enumeration a framework using gro bner basis reductions in this paper we present a framework for solving stochastic programs with complete integer recourse and discretely distributed right hand side vector using grobner basis methods from computational algebra to solve the numerous second stage integer programs using structural properties of the expected integer recourse function we prove that under mild conditions an optimal solution is contained in a finite set furthermore we present a basic scheme to enumerate this set and suggest improvements to reduce the number of function evaluations needed

a cutting plane approach to mixed 0 1 stochastic integer programs abstract we consider a mixed 0 1 integer programming problem with dual block angular structure arising in two stage stochastic programming a relaxation is proposed such that the problem is decomposed into subproblems each corresponding to the outcomes of the random variable the convex hull of feasible solutions for the relaxation is characterized using results from disjunctive programming and it is shown how lift and project cuts can be generated for one subproblem and made valid for different outcomes

a closed form representation of mixed integer program value functions we identify a class of formulas computable in polynomial time such that the functions defined by these formulas are precisely the value functions of mixed integer programs with rational constraint coefficients

sufficient conditions for a nondegenerate hopf bifurcation in a generalized lagrange poisson problem in this paper we provide sufficient conditions for the existence of a nondegenerate hamiltonian hopf bifurcation at the equilibria corresponding to the rotation around the vertical axis of a symmetric gyrostat with a fixed point under the effect of an axially symmetric potential

on well posedness of the muskat problem with surface tension abstract we consider the muskat problem with surface tension for one fluid or two fluids with or without viscosity jump with infinite depth or lipschitz rigid boundaries and in arbitrary dimension d of the interface the problem is nonlocal quasilinear and to leading order is scaling invariant in the sobolev space h s c r d with s c 1 d 2 we prove local well posedness for large data in all subcritical sobolev spaces h s r d s s c allowing for initial interfaces whose curvatures are unbounded and furthermore when d 1 not locally square integrable to the best of our knowledge this is the first large data well posedness result that covers all subcritical sobolev spaces for the muskat problem with surface tension we reformulate the problem in terms of the dirichlet neumann operator and use a paradifferential approach to reduce the problem to an explicit parabolic equation which is of independent interest

scenarios and policy aggregation in optimization under uncertainty a common approach in coping with multiperiod optimization problems under uncertainty where statistical information is not really enough to support a stochastic programming model has been to set up and analyze a number of scenarios the aim then is to identify trends and essential features on which a robust decision policy can be based this paper develops for the first time a rigorous algorithmic procedure for determining such a policy in response to any weighting of the scenarios the scenarios are bundled at various levels to reflect the availability of information and iterative adjustments are made to the decision policy to adapt to this structure and remove the dependence on hindsight

the value function of an integer program we consider integer programs in which the objective function and constraint matrix are fixed while the right hand side varies the value function gives for each feasible right hand side the criterion value of the optimal solution we provide a precise characterization of the closed form expression for any value function

analysis and approximation of nonlocal diffusion problems with volume constraints a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in mbrn the nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion including a notion of nonlocal flux the ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory in particular it is shown that fractional laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider the numerous applications elucidate different interpretations of the operator and the associated governing equations for example a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process sufficie

a cutting plane approach to capacitated lot sizing with start up costs we consider a mixed integer model for multi item single machine production planning incorporating both start up costs and machine capacity the single item version of this model is studied from the polyhedral point of view and several families of valid inequalities are derived for some of these inequalities we give necessary and sufficient facet inducing conditions and efficient separation algorithms we use these inequalities in a cutting plane branch and bound procedure a set of real life based problems with 5 items and up to 36 periods is solved to optimality

lower bounds in lot sizing models a polyhedral study variable lower bounds in mixed integer programs are constraints with the general form x ly where xis a continuous variable and y is a binary or an integer variable this type of constraints is present in some lot sizing models where x represents the amount of some article produced by a machine and y 0 1 indicates whether the machine is set up for that articley 1 or not y 0 in these models production below some level is not allowed in order to make full use of resources in this paper we study from the polyhedral viewpoint the mixed integer models whose feasible set includes an additive constraint xj d and constraints lyj xj kyj where xj are continuous variables yj are integer or binary variables and k is a large constant we derive families of strong valid inequalities and show they are sufficient to describe completely the convex hulls of the sets of feasible solutions moreover we develop polynomial algorithms to solve the separation problem associated to each of the families obtained finally we incorporate the inequalities obtained in a cutting plane algorithm and test their ability to solve some multi item lot sizing problems

tight mip formulations for bounded up down times and interval dependent start ups switching machines on and off is an important aspect of unit commitment problems and production planning problems among others here we study tight mixed integer programming formulations for two aspects of such problems bounded length on and off intervals and interval dependent start ups the problem with both these aspects admits a general dynamic programming shortest path formulation which leads to a tight extended formulation with a number of binary variables that is quadratic in the number n of time periods we are thus interested in tight formulations with a linear number of binary variables for the bounded interval problem we present a tight network dual formulation based on new integer cumulative variables that allows us to simultaneously treat lower and upper bounds on the interval lengths and also to handle time varying bounds this in turn leads to more general results including simpler proofs of known tight formulations for problems with just lower bounds for the interval dependent start up problem we develop a path formulation that allows us to describe the convex hull of solutions in the space of machine state variables and interval dependent start up variables

equilibria stability and hamiltonian hopf bifurcation of a gyrostat in an incompressible ideal fluid abstract for a gyrostat in a incompressible ideal fluid by writing kirchhoff s equations as a lie poisson system and using a non canonical hamiltonian formulation we provide the expressions of the equilibria when the gyrostatic momentum is constant with the form l 0 0 l and present necessary and sufficient conditions for the stability of some of them via the energy casimir method and the study of the linearized equations of the motion finally using a recent geometric method introduced by hanssmann and van der meer we give a sufficient condition for the existence of a non degenerate hamiltonian hopf bifurcation at those equilibria when the gyrostat is symmetric

lifting inequalities a framework for generating strong cuts for nonlinear programs in this paper we introduce the first generic lifting techniques for deriving strong globally valid cuts for nonlinear programs the theory is geometric and provides insights into lifting based cut generation procedures yielding short proofs of earlier results in mixed integer programming using convex extensions we obtain conditions that allow for sequence independent lifting in nonlinear settings paving a way for efficient cut generation procedures for nonlinear programs this sequence independent lifting framework also subsumes the superadditive lifting theory that has been used to generate many general purpose strong cuts for integer programs we specialize our lifting results to derive facet defining inequalities for mixed integer bilinear knapsack sets finally we demonstrate the strength of nonlinear lifting by showing that these inequalities cannot be obtained using a single round of traditional integer programming cut generation techniques applied on a tight reformulation of the problem

a branch and cut algorithm for the stochastic uncapacitated lot sizing problem this paper addresses a multi stage stochastic integer programming formulation of the uncapacitated lot sizing problem under uncertainty we show that the classical l s inequalities for the deterministic lot sizing polytope are also valid for the stochastic lot sizing polytope we then extend the l s inequalities to a general class of valid inequalities called the inequalities and we establish necessary and sufficient conditions which guarantee that the inequalities are facet defining a separation heuristic for inequalities is developed and incorporated into a branch and cut algorithm a computational study verifies the usefulness of the inequalities as cuts

a branch and price algorithm for multistage stochastic integer programming with application to stochastic batch sizing problems in this paper we present a branch and price method to solve special structured multistage stochastic integer programming problems we validate our method on two different versions of a multistage stochastic batch sizing problem sbsp one version adopts a recourse formulation and the other is based on probabilistic constraints our algorithmic approach is applicable to both formulations our computational results suggest that both classes of problems can be solved using relatively few nodes of a branch and price tree the success of our approach calls for extensions in methodology as well as applications

lagrangian relative equilibria for a gyrostat in the three body problem bifurcations and stability in this paper we consider the non canonical hamiltonian dynamics of a gyrostat in the frame of the three body problem using geometric mechanic methods we study the approximate dynamics of the truncated legendre series representation of the potential of an arbitrary order working in the reduced problem we study the existence of relative equilibria that we refer to as lagrange type following the analogy with the standard techniques we provide necessary and sufficient conditions for the linear stability of lagrangian relative equilibria if the gyrostat morphology form is close to a sphere thus we generalize the classical results on equilibria of the three body problem and many results on them obtained by the classic approach for the case of rigid bodies

cutting planes for the multistage stochastic unit commitment problem as renewable energy penetration rates continue to increase in power systems worldwide new challenges arise for system operators in both regulated and deregulated electricity markets to solve the security constrained coal fired unit commitment problem with intermittent generation due to renewables and uncertain load in order to ensure system reliability and maintain cost effectiveness in this paper we study a security constrained coal fired stochastic unit commitment model which we use to enhance the reliability unit commitment process for day ahead power system operations in our approach we first develop a deterministic equivalent formulation for the problem which leads to a large scale mixed integer linear program then we verify that the turn on off inequalities provide a convex hull representation of the minimum up down time polytope under the stochastic setting next we develop several families of strong valid inequalities mainly through lifting schemes in particular by exploring sequence independent lifting and subadditive approximation lifting properties for the lifting schemes we obtain strong valid inequalities for the ramping and general load balance polytopes finally branch and cut algorithms are developed to employ these valid inequalities as cutting planes to solve the problem our computational results verify the effectiveness of the proposed approach