a branch and cut decomposition algorithm for solving chance constrained mathematical programs with finite support we present a new approach for exactly solving chance constrained mathematical programs having discrete distributions with finite support and random polyhedral constraints such problems have been notoriously difficult to solve due to nonconvexity of the feasible region and most available methods are only able to find provably good solutions in certain very special cases our approach uses both decomposition to enable processing subproblems corresponding to one possible outcome at a time and integer programming techniques to combine the results of these subproblems to yield strong valid inequalities computational results on a chance constrained formulation of a resource planning problem inspired by a call center staffing application indicate the approach works significantly better than both an existing mixed integer programming formulation and a simple decomposition approach that does not use strong valid inequalities we also demonstrate how the approach can be used to efficiently solve for a sequence of risk levels as would be done when solving for the efficient frontier of risk and cost

the complexity of mean payoff games on graphs abstract we study the complexity of finding the values and optimal strategies of mean payoff games on graphs a family of perfect information games introduced by ehrenfeucht and mycielski and considered by gurvich karzanov and khachiyan we describe a pseudo polynomial time algorithm for the solution of such games the decision problem for which is in np conp finally we describe a polynomial reduction from mean payoff games to the simple stochastic games studied by condon these games are also known to be in np conp but no polynomial or pseudo polynomial time algorithm is known for them

the sample average approximation method for stochastic discrete optimization in this paper we study a monte carlo simulation based approach to stochastic discrete optimization problems the basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function the obtained sample average optimization problem is solved and the procedure is repeated several times until a stopping criterion is satisfied we discuss convergence rates stopping rules and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem

improved handling of uncertainty and robustness in set covering problems abstract set covering problems are one of the classical combinatorial optimization problems and part of many different practical applications e g the location of emergency service facilities in many real world problems the required statistical parameters are not precisely known and the obtained solutions may reveal a non adequate performance we introduce a robust formulation of the uncertain probabilistic set covering problem which combines the concepts of robust and probabilistic optimization by introducing robust covering constraints this robust uncertain set covering problem can be stated as a compact mixed integer linear programming model additionally two non compact integer linear model formulations are developed as the strength of these formulations is not known a priori we analyze the performance of these formulations in an extensive computational study a case study for the location of emergency service facilities highlights the benefits of our approach in comparison to a formulation neglecting these uncertainties

duality gaps in nonconvex stochastic optimization we consider multistage stochastic optimization models containing nonconvex constraints e g due to logical or integrality requirements we study three variants of lagrangian relaxations and of the corresponding decomposition schemes namely scenario nodal and geographical decomposition based on convex equivalents for the lagrangian duals we compare the duality gaps for these decomposition schemes the first main result states that scenario decomposition provides a smaller or equal duality gap than nodal decomposition the second group of results concerns large stochastic optimization models with loosely coupled components the results provide conditions implying relations between the duality gaps of geographical decomposition and the duality gaps for scenario and nodal decomposition respectively

convex proximal bundle methods in depth a unified analysis for inexact oracles the last few years have seen the advent of a new generation of bundle methods capable to handle inexact oracles polluted by noise proving convergence of a bundle method is never simple and coping with inexact oracles substantially increases the technicalities besides several variants exist to deal with noise each one needing an ad hoc proof to show convergence we state a synthetic convergence theory in which we highlight the main arguments and specify which assumption is used to establish each intermediate result the framework is comprehensive and generalizes in various ways a number of algorithms proposed in the literature based on the ingredients of our synthetic theory we consider various bundle methods adapted to oracles for which high accuracy is possible yet it is preferable not to make exact calculations often because they are too time consuming

relaxations and approximations of chance constraints under finite distributions optimization problems with constraints involving stochastic parameters that are required to be satisfied with a prespecified probability threshold arise in numerous applications such chance constrained optimization problems involve the dual challenges of stochasticity and nonconvexity in the setting of a finite distribution of the stochastic parameters an optimization problem with linear chance constraints can be formulated as a mixed integer linear program milp the natural milp formulation has a weak relaxation bound and is quite difficult to solve in this paper we review some recent results on improving the relaxation bounds and constructing approximate solutions for milp formulations of chance constraints we also discuss a recently introduced bicriteria approximation algorithm for covering type chance constrained problems this algorithm uses a relaxation to construct a solution whose constraint violation risk level may be larger than the pre specified threshold but is within a constant factor of it and whose objective value is also within a constant factor of the true optimal value finally we present some new results that improve on the bicriteria approximation factors in the finite scenario setting and shed light on the effect of strong relaxations on the approximation ratios

a convergent numerical scheme for integrodifferential kinetic models of angiogenesis abstract we study a robust finite difference scheme for integrodifferential kinetic systems of fokker planck type modeling tumor driven blood vessel growth the scheme is of order one and enjoys positivity features we analyze stability and convergence properties and show that soliton like asymptotic solutions are correctly captured we also find good agreement with the solution of the original stochastic model from which the deterministic kinetic equations are derived working with ensemble averages a numerical study clarifies the influence of velocity cut offs on the solutions for exponentially decaying data

on quantile cuts and their closure for chance constrained optimization problems a chance constrained optimization problem over a finite distribution involves a set of scenario constraints from which a small subset can be violated we consider the setting where all scenario constraints are mixed integer convex existing works typically consider a mixed integer nonlinear programming minlp formulation of this problem by introducing binary variables to indicate which constraint systems are to be satisfied or violated a variety of cutting plane approaches for this minlp formulation have been developed in this paper we consider a family of cuts in the original space rather than those in the extended space of the minlp reformulation these cuts known as quantile cuts can be viewed as a projection of the well known family of mixing inequalities for the minlp reformulation onto the original problem space we show that the closure of the infinite family of all quantile cuts has a finite description an important corollary of this result is that for linear chance constrained problems the quantile closure is polyhedral we further show that a recursive application of quantile closure operations recovers the convex hull of the nonconvex chance constrained set in the limit and in the pure integer setting the convergence is finite we show that separation of quantile cuts is in general np hard develop a heuristic separation method and demonstrate its effectiveness through a computational study we also study an approximation of the quantile closure and propose a generalization by grouping scenarios

decomposition algorithms for two stage chance constrained programs we study a class of chance constrained two stage stochastic optimization problems where second stage feasible recourse decisions incur additional cost in addition we propose a new model where recovery decisions are made for the infeasible scenarios to obtain feasible solutions to a relaxed second stage problem we develop decomposition algorithms with specialized optimality and feasibility cuts to solve this class of problems computational results on a chance constrained resource planing problem indicate that our algorithms are highly effective in solving these problems compared to a mixed integer programming reformulation and a naive decomposition method

abstract tubes improved inclusion exclusion identities and inequalities and importance sampling numerous statistical applications require the evaluation of the probability content of a convex polyhedron we demonstrate for a given polyhedron in r d that there is a depth d inclusion exclusion identity for its indicator function which is a linear combination of indicator functions of intersections of at most d half spaces terms in the identity are determined by the incidence of the facets of the polyhedron which can be found using linear programming this identity can be truncated at any depth to give a lower or upper bound in addition the resulting inequalities lead to importance sampling schemes for evaluating the probability content and these methods tend to be more efficient than the naive hit or miss monte carlo method these results arise in a more general setting which we introduce an tube consists of a pair a s where a a1 a n is a collection of sets s is a simplicial complex and where each subcomplex s x f s x i f a i is contractible whenever x u i n 1 a i the notion presented here is stronger than the one introduced earlier by naiman and wynn several examples are given and key consequences are demonstrated in particular arrangements of points and half spaces in r d give rise to abstract tubes via voronoi decompositions and their associated delauney dual complexes every abstract tube is shown to give rise to an inclusion exclusion identity for i u n i 1 a i and upper and lower bounds are obtained by truncating the identity at an even or an odd depth this property is analogous to the truncation inequality property of the classical inclusion exclusion identity which may be viewed as a special case the notion of an abstract subtube is introduced and it is shown that if a s 1 is a subtube of a s 2 then the truncation inequality gotten from the depth m truncation for a s 1 is at least as sharp as the corresponding inequality from a s 2 as a consequence the generalized inclusion exclusion inequalities are always at least as sharp as their classical counterparts

probabilistic set covering with correlations we consider two variants of a probabilistic set covering psc problem the first variant assumes that there is uncertainty regarding whether a selected set can cover an item and the objective is to determine a minimum cost combination of sets so that each item is covered with a prespecified probability the second variant seeks to maximize the minimum probability that a selected set can cover all items to date literature on this problem has focused on the special case in which uncertainties are independent in this paper we formulate deterministic mixed integer programming models for distributionally robust psc problems with correlated uncertainties by exploiting the supermodularity of certain substructures and analyzing their polyhedral properties we develop strong valid inequalities to strengthen the formulations computational results illustrate that our modeling approach can outperform formulations in which correlations are ignored and that our algorithms can significantly reduce overall comput

hybrid modeling of tumor induced angiogenesis when modeling of tumor driven angiogenesis a major source of analytical and computational complexity is the strong coupling between the kinetic parameters of the relevant stochastic branching and growth of the capillary network and the family of interacting underlying fields to reduce this complexity we take advantage of the system intrinsic multiscale structure we describe the stochastic dynamics of the cells at the vessel tip at their natural mesoscale whereas we describe the deterministic dynamics of the underlying fields at a larger macroscale here we set up a conceptual stochastic model including branching elongation and anastomosis of vessels and derive a mean field approximation for their densities this leads to a deterministic integropartial differential system that describes the formation of the stochastic vessel network we discuss the proper capillary injecting boundary conditions and include the results of relevant numerical simulations

on the separation of disjunctive cuts disjunctive cuts for mixed integer linear programs mips were introduced by egon balas in the late 1970s and have been successfully exploited in practice since the late 1990s in this paper we investigate the main ingredients of a disjunctive cut separation procedure and analyze their impact on the quality of the root node bound for a set of instances taken from miplib library we compare alternative normalization conditions and try to better understand their role in particular we point out that constraints that become redundant because of the disjunction used can produce over weak cuts and analyze this property with respect to the normalization used finally we introduce a new normalization condition and analyze its theoretical properties and computational behavior along the way we make use of a number of small numerical examples to illustrate some basic and often misinterpreted disjunctive programming features

asymptotic formulas for the derivatives of probability functions and their monte carlo estimations one of the key problems in chance constrained programming for nonlinear optimization problems is the evaluation of derivatives of joint probability functions of the form p x p gp x k 6 cp p 1 nc here x 2 r nx is the vector of physical parameters k 2 r nk is a random vector describing the uncertainty of the model g r nx r nk r nc is the constraints mapping and c 2 r nc is the vector of constraint levels in this paper specific monte carlo tools for the estimations of the gradient and hessian of p x are proposed when the input random vector k has a multivariate normal distribution and small variances using the small variance hypothesis approximate expressions for the first and second order derivatives are obtained whose monte carlo estimations have low computational costs the number of calls of the constraints mapping g for the proposed estimators of the gradient and hessian of p x is only 1 2nx 2nk these tools are implemented in penalized optimization routines adapted to stochastic optimization and are shown to reduce the computational cost of chance constrained programming substantially

n by n checkers is exptime complete the game of checkers can easily be generalized to be played on an n by n board and the complexity of deciding questions about positions regarded as a function of n this paper considers mainly the question of whether a particular player can force a win from a given position and also the question of what is the best move in a given position each of these problems is shown to be complete in exponential time this means that any algorithm to solve them must take time which rises exponentially with respect to some power of n and moreover that they are amongst the hardest problems with such a time bound for instance if there are any problems solvable in exponential time but not in polynomial space then these two problems are amongst them

solitonlike attractor for blood vessel tip density in angiogenesis recently numerical simulations of a stochastic model have shown that the density of vessel tips in tumor induced angiogenesis adopts a solitonlike profile sci rep 6 31296 2016 2045 232210 1038 srep31296 in this work we derive and solve the equations for the soliton collective coordinates that indicate how the soliton adapts its shape and velocity to varying chemotaxis and diffusion the vessel tip density can be reconstructed from the soliton formulas while the stochastic model exhibits large fluctuations we show that the location of the maximum vessel tip density for different replicas follows closely the soliton peak position calculated either by ensemble averages or by solving an alternative deterministic description of the density the simple soliton collective coordinate equations may also be used to ascertain the response of the vessel network to changes in the parameters and thus to control it

regularization methods for optimization problems with probabilistic constraints we analyze nonlinear stochastic optimization problems with probabilistic constraints on nonlinear inequalities with random right hand sides we develop two numerical methods with regularization for their numerical solution the methods are based on first order optimality conditions and successive inner approximations of the feasible set by progressive generation of p efficient points the algorithms yield an optimal solution for problems involving concave probability distributions for arbitrary distributions the algorithms solve the convex hull problem and provide upper and lower bounds for the optimal value and nearly optimal solutions the methods are compared numerically to two cutting plane methods

alternation with restrictions on looping alternating turing machines with restrictions preventing them from returning to a previous configuration model games with rules enforcing such a restriction for instance the chinese version of go such restrictions do not affect the time complexity of problems for alternating turing machines but space s on a machine with the restriction is equivalent either to time or to space exponential in s on a normal alternating machine depending on the precise nature of the restriction

recent results and questions in combinatorial game complexities abstract aim present a systematic development of part of the theory of combinatorial games from the ground up approach computational complexity combinatorial games are completely determined the questions of interest are efficiencies of strategies methodology divide and conquer isolate the various difficulties separating hard from easy games and attack them individually presentation informal examples of games sampled from various levels illustrate the theory with emphasis on formulating and motivating new and old research problems

lower bounds for multiplayer noncooperative games of incomplete information abstract this paper see also 1 extends the alternating turing machine a tm of chandra kozen and stockmeyer 2 the private and the blind alternating machines of reif 3 4 to model multiplayer games of incomplete information we use these machines to provide matching lower bounds for our decision algorithms described in our companion paper 5 we also apply multiple person alternation to other machine types we show that multiplayer games of incomplete information can be undecidable general however one form of incomplete information games that is decidable we term as hierarchical games defined later in this paper in hierarchical multiplayer games each additional clique subset of players with same information increases the complexity of the outcome problem by a further exponential consequently if a multiplayer game of incomplete information with k cliques has a space bound of s n then its outcome can be k repeated exponentials harder than games of complete information with the same space bound s n this paper proves that this exponential blow up must occur in the worst case we define team private peek and team blind peek extending the previous models of peek these new games can be shown to be complete for their respective classes we use these games to establish lower bounds on complexity of multiplayer games of incomplete information and blindfold multiplayer games we analyze the time bounded alternating machines and conclude that time is not a very critical resource for multiplayer alternation we also show dqbf a variant of qbf to be complete in nexptime

undirected edge geography abstract the game of edge geography is played by two players who alternately move a token on a graph from one vertex to an adjacent vertex erasing the edge in between the player who first has no legal move is the loser we show that the decision problem of determining whether a position in this game is a win for the first player is pspace complete further the problem remains pspace complete when restricted to planar graphs with maximum degree 3 however if the underlying graph is bipartite we provide 1 a linear algebraic characterization of the p and n positions yielding 2 a polynomial time algorithm for deciding whether any given position is p or n and also 3 a polynomial time algorithm to find winning moves

bicriteria approximation of chance constrained covering problems a chance constrained optimization problem involves constraints with random data that can be violated with probability bounded from above by a prespecified small risk parameter such constraints are

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

pattern based modeling and solution of probabilistically constrained optimization problems we propose a new modeling and solution method for probabilistically constrained optimization problems the methodology is based on the integration of the stochastic programming and combinatorial pattern recognition fields it permits the fast solution of stochastic optimization problems in which the random variables are represented by an extremely large number of scenarios the method involves the binarization of the probability distribution and the generation of a consistent partially defined boolean function pdbf representing the combination f p of the binarized probability distribution f and the enforced probability level p we show that the pdbf representing f p can be compactly extended as a disjunctive normal form dnf the dnf is a collection of combinatorial p patterns each defining sufficient conditions for a probabilistic constraint to hold we propose two linear programming formulations for the generation of p patterns that can be subsequently used to derive a linear programming inner approximation of the original stochastic problem a formulation allowing for the concurrent generation of a p pattern and the solution of the deterministic equivalent of the stochastic problem is also proposed the number of binary variables included in the deterministic equivalent formulation is not an increasing function of the number of scenarios used to represent uncertainty results show that large scale stochastic problems in which up to 50 000 scenarios are used to describe the stochastic variables can be consistently solved to optimality within a few seconds

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

data driven chance constrained stochastic program in this paper we study data driven chance constrained stochastic programs or more specifically stochastic programs with distributionally robust chance constraints dccs in a data driven setting to provide robust solutions for the classical chance constrained stochastic program facing ambiguous probability distributions of random parameters we consider a family of density based confidence sets based on a general phi divergence measure and formulate dcc from the perspective of robust feasibility by allowing the ambiguous distribution to run adversely within its confidence set we derive an equivalent reformulation for dcc and show that it is equivalent to a classical chance constraint with a perturbed risk level we also show how to evaluate the perturbed risk level by using a bisection line search algorithm for general phi divergence measures in several special cases our results can be strengthened such that we can derive closed form expressions for the perturbed risk levels in addition we show that the conservatism of dcc vanishes as the size of historical data goes to infinity furthermore we analyze the relationship between the conservatism of dcc and the size of historical data which can help indicate the value of data finally we conduct extensive computational experiments to test the performance of the proposed dcc model and compare various phi divergence measures based on a capacitated lot sizing problem with a quality of service requirement

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

the othello game on an n n board is pspace complete abstract given an arbitrary position of the othello game played on an n n board the problem of determining the winner is shown to be pspace complete it can be reduced from generalized geography played on bipartite graphs with maximum degree 3

optimized bonferroni approximations of distributionally robust joint chance constraints a distributionally robust joint chance constraint involves a set of uncertain linear inequalities which can be violated up to a given probability threshold epsilon over a given family of probability distributions of the uncertain parameters a conservative approximation of a joint chance constraint often referred to as a bonferroni approximation uses the union bound to approximate the joint chance constraint by a system of single chance constraints one for each original uncertain constraint for a fixed choice of violation probabilities of the single chance constraints such that their sum does not exceed epsilon it has been shown that under various settings a distributionally robust single chance constraint admits a deterministic convex reformulation thus the bonferroni approximation approach can be used to build convex approximations of distributionally robust joint chance constraints in this paper we consider an optimized version of bonferroni approximation where the violation probabilities of the individual single chance constraints are design variables rather than fixed a priori we show that such an optimized bonferroni approximation of a distributionally robust joint chance constraint is exact when the uncertainties are separable across the individual inequalities i e each uncertain constraint involves a different set of uncertain parameters and corresponding distribution families unfortunately the optimized bonferroni approximation leads to np hard optimization problems even in settings where the usual bonferroni approximation is tractable when the distribution family is specified by moments or by marginal distributions we derive various sufficient conditions under which the optimized bonferroni approximation is convex and tractable we also show that for moment based distribution families and binary decision variables the optimized bonferroni approximation can be reformulated as a mixed integer second order conic set finally we demonstrate how our results can be used to derive a convex reformulation of a distributionally robust joint chance constraint with a specific non separable distribution family

ambiguous risk constraints with moment and unimodality information optimization problems face random constraint violations when uncertainty arises in constraint parameters effective ways of controlling such violations include risk constraints e g chance constraints and conditional value at risk constraints this paper studies these two types of risk constraints when the probability distribution of the uncertain parameters is ambiguous in particular we assume that the distributional information consists of the first two moments of the uncertainty and a generalized notion of unimodality we find that the ambiguous risk constraints in this setting can be recast as a set of second order cone soc constraints in order to facilitate the algorithmic implementation we also derive efficient ways of finding violated soc constraints finally we demonstrate the theoretical results via computational case studies on power system operations

a geometric study of duality gaps with applications lagrangian relaxation is often an efficient tool to solve large scale optimization problems even nonconvex however it introduces a duality gap which should be small for the method to be really efficient here we make a geometric study of the duality gap given a nonconvex problem we formulate in a first part a convex problem having the same dual this formulation involves a convexification in the product of the three spaces containing respectively the variables the objective and the constraints we apply our results to several relaxation schemes especially one called lagrangean decomposition in the combinatorial optimization community or operator splitting elsewhere we also study a specific application highly nonlinear the unit commitment problem

an algorithmic analysis of the honey bee game the honey bee game is a two player board game that is played on a connected hexagonal colored grid or in a generalized setting on a connected graph with colored nodes in a single move a player calls a color and thereby conquers all the nodes of that color that are adjacent to his own current territory both players want to conquer the majority of the nodes we show that winning the game is pspace hard in general np hard on series parallel graphs but easy on outer planar graphs in the solitaire version the goal of the single player is to conquer the entire graph with the minimum number of moves the solitaire version is np hard on trees and split graphs but can be solved in polynomial time on co comparability graphs

a proximal bundle method with approximate subgradient linearizations we give a proximal bundle method for minimizing a convex function f over a closed convex set it only requires evaluating f and its subgradients with an accuracy epsilon 0 which is fixed but possibly unknown it asymptotically finds points that are epsilon optimal when applied to lagrangian relaxation it allows for epsilon accurate solutions of lagrangian subproblems and finds epsilon optimal solutions of convex programs

complexity of path forming games abstract for a number of two player games where players alternately choose the next vertex of a simple or an elementary path in a graph we consider the problem to determine whether for a given game instance there is a winning strategy for the first player we show several of these problems to be pspace complete in some special cases we obtain polynomial time algorithms based on graph rewriting or an intricate form of dynamic programming e g we show generalized geography and some other pspace complete problems to be linear time solvable on graphs with constant bounded treewidth

phutball is pspace hard we consider the nxn game of phutball it is shown that given an arbitrary position of stones on the board it is a pspace hard problem to determine whether the specified player can win the game regardless of the opponent s choices made during the game

the probabilistic set covering problem in a probabilistic set covering problem the right hand side is a random binary vector and the covering constraint has to be satisfied with some prescribed probability we analyze the structure of the set of probabilistically efficient points of binary random vectors develop methods for their enumeration and propose specialized branch and bound algorithms for probabilistic set covering problems

a gradient formula for linear chance constraints under gaussian distribution we provide an explicit gradient formula for linear chance constraints under a possibly singular multivariate gaussian distribution this formula allows one to reduce the calculus of gradients to the calculus of values of the same type of chance constraints in smaller dimension and with different distribution parameters this is an important aspect for the numerical solution of stochastic optimization problems because existing efficient codes for e g calculating singular gaussian distributions or regular gaussian probabilities of polyhedra can be employed to calculate gradients at the same time moreover the precision of gradients can be controlled by that of function values which is a great advantage over using finite difference approximations finally higher order derivatives are easily derived explicitly the use of the obtained formula is illustrated for an example of a transportation network with stochastic demands

differentiation of probability functions the transformation method abstract in reliability oriented design and optimization of engineering systems one needs various derivatives of the probability of systems survival p x p yl y a w x yu here y y a x denotes the vector of response or output variables depending on the design or input vector x and the vector of random system parameters a a we assume that a has a given probability density function a furthermore y l y u are the vectors of given lower and upper bounds for y there is shown that in many cases derivatives d l p x of arbitrary order l can be obtained by applying an integral transformation t x to the integral representation of p x such that the transformed domain of integration becomes independent of x the derivatives result then by interchanging differentiation and integration based on the mean value representations of d l p x found in the first part estimations of the derivatives can be obtained by using several sampling techniques furthermore having the mentioned integral representations d l p x can be computed approximately by writing d l p x first as a laplace integral and applying then the asymptotic expansion techniques known for laplace integrals

constrained bundle methods for upper inexact oracles with application to joint chance constrained energy problems joint chance constrained problems give rise to many algorithmic challenges even in the convex case i e when an appropriate transformation of the probabilistic constraint is a convex function its cutting plane linearization is just an approximation produced by an oracle providing subgradient and function values that can only be evaluated inexactly as a result the cutting plane model may lie above the true constraint for dealing with such upper inexact oracles and still solving the problem up to certain precision a special numerical algorithm must be put in place we introduce a family of constrained bundle methods based on the so called improvement functions that is shown to be convergent and encompasses many previous approaches as well as new algorithms depending on the oracle accuracy we analyze to what extent the considered methods solve the joint chance constrained program the approach is assessed on real life energy problems arising when dealing with stochastic hydroreservoir management

on the fairness and complexity of generalized k in a row games recently wu and huang i c wu d y huang a new family of k in a row games in the 11th advances in computer games conference acg 11 taipei taiwan september 2005 introduced a new game called connect6 where two players black and white alternately place two stones of their own color black and white respectively on an empty go like board except for that black the first player places one stone only for the first move the one who gets six consecutive horizontally vertically or diagonally stones of his color first wins the game unlike go moku connect6 appears to be fairer and has been adopted as an official competition event in computer olympiad 2006 connect m n k p q is a generalized family of k in a row games where two players place p stones on an mxn board alternatively except black places q stones in the first move the one who first gets his stones k consecutive in a line horizontally vertically or diagonally wins connect6 is simply the game of connect m n 6 2 1 in this paper we study two interesting issues of connect m n k p q fairness and complexity first we prove that no one has a winning strategy in connect m n k p q starting from an empty board when k 4p 7 and p q second we prove that for any fixed constants k p such that k p max 3 p and a given connect m n k p q position it is pspace complete to determine whether the first player has a winning strategy consequently this implies that connect6 played on an mxn board i e connect m n 6 2 1 is pspace complete

complexity of path discovery game problems in this paper path discovery games are introduced and complexity of the game problems is studied it is shown that the path discovery game problem played on directed graphs is pspace complete and the path discovery game problem played on undirected graphs is in the class sspace nlogn moreover it is shown that the acyclic path discovery game problems played on directed graphs and on undirected graphs are both np complete

all structured programs have small tree width and good register allocation the register allocation problem for an imperative program is often modelled as the coloring problem of the interference graph of the control flow graph of the program the interference graph of a flow graph g is the intersection graph of some connected subgraphs of g these connected subgraphs represent the lives or life times of variables so the coloring problem models that two variables with overlapping life times should be in different registers for general programs with unrestricted gotos the interference graph can be any graph and hence we cannot in general color within a factor o n from optimality unless np p

ambiguous chance constrained problems and robust optimization in this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain we focus primarily on the special case where the uncertainty set of the distributions is of the form where p denotes the prohorov metric the ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure the main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with a high probability this result is established using the strassen dudley representation theorem that states that when the distributions of two random variables are close in the prohorov metric one can construct a coupling of the random variables such that the samples are close with a high probability we also show that the robust sampled problem can be solved efficiently both in theory and in practice

a finite set of functions with an exptime complete composition problem we exhibit a finite family of functions over a finite set i e a finite algebra such that the problem whether a given function can be obtained as a composition of the members of this family i e is a member of the clone generated by the algebra is exptime complete

the monadic second order logic of graphs i recognizable sets of finite graphs abstract the notion of a recognizable set of finite graphs is introduced every set of finite graphs that is definable in monadic second order logic is recognizable but not vice versa the monadic second order theory of a context free set of graphs is decidable

the size of bdds and other data structures in temporal logics model checking temporal logic model checking is a verification method in which we describe a system the model and then we verify whether important properties expressed in a temporal logic formula hold in the system many model checking tools employ bdds or some other data structure to represent sets of states it has been empirically observed that the bdds used in these algorithms may grow exponentially as the model and formula increase in size we formally prove that no kind of data structure of polynomial size can represent the set of valid initial states for all models and all formulae this result holds for all data structures where a state can be checked in polynomial time therefore it holds not only for all types of bdds regardless of variable ordering but also for more powerful data structures such as rbcs mtbdds adds and sdds thus the size explosion of bdds is not a limit of these specific data representation structures but is unavoidable every formalism used in the same way would lead to an exponential size blow up

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

probability gradient estimation by set valued calculus and applications in network design let vartheta mapsto p vartheta be a set valued mapping from rr d into the family of closed compact polyhedra in rr s let xi be a rr s valued random variable many stochastic optimization problems in computer networking system reliability transportation telecommunication finance etc can be formulated as a problem to minimize or maximize the probability pp xi in p vartheta under some constraints on the decision variable vartheta for a practical solution of such a problem one has to approximate the objective function and its derivative by monte carlo simulation since a closed analytical expression is only available in rare cases in this paper we present a new method of approximating the gradient of pp xi in p vartheta w r t vartheta by sampling which is based on the concept of setwise weak derivative quite surprisingly it turns out that it is typically easier to approximate the derivative than the objective itself

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

monadic second order evaluations on tree decomposable graphs abstract every graph generated by a hyperedge replacement graph grammar can be represented by a tree namely the derivation tree of the derivation sequence that produced it certain functions on graphs can be computed recursively on the derivation trees of these graphs by using monadic second order logic and semiring homomorphisms we describe in a single formalism a large class of such functions polynomial and even linear algorithms can be constructed for some of these functions we unify similar results obtained by takamizawa 1982 bern 1987 arnborg 1991 and habel 1989

solving chance constrained optimization problems with stochastic quadratic inequalities we propose a new and systematic reformulation and algorithmic approach to solve a complex class of stochastic programming problems involving a joint chance constraint with random technology matrix and stochastic quadratic inequalities the method is general enough to apply to nonconvex as well as nonseparable quadratic terms we derive two new reformulations and give sufficient conditions under which the reformulated problem is equivalent the second reformulation provides a much sparser representation of the feasible set of the chance constraint and offers tremendous computational advantages this new reformulation method can be used for linear stochastic inequalities and will also significantly improve the solution of such joint chance constrained problems we provide general and easily identifiable conditions under which the base reformulations can be linearized we show that the size of the reformulated problems in particular their number of binary variables and quadratic mixed integer terms does n

on mixing sets arising in chance constrained programming the mixing set with a knapsack constraint arises in deterministic equivalent of chance constrained programming problems with finite discrete distributions we first consider the case that the chance constrained program has equal probabilities for each scenario we study the resulting mixing set with a cardinality constraint and propose facet defining inequalities that subsume known explicit inequalities for this set we extend these inequalities to obtain valid inequalities for the mixing set with a knapsack constraint in addition we propose a compact extended reformulation with polynomial number of variables and constraints that characterizes a linear programming equivalent of a single chance constraint with equal scenario probabilities we introduce a blending procedure to find valid inequalities for intersection of multiple mixing sets we propose a polynomial size extended formulation for the intersection of multiple mixing sets with a knapsack constraint that is stronger than the original mixing formulation we also give a compact extended linear program for the intersection of multiple mixing sets and a cardinality constraint for a special case we illustrate the effectiveness of the proposed inequalities in our computational experiments with probabilistic lot sizing problems

efficient solution to connectivity problems on hierarchically defined graphs using hierarchical definitions we can describe very large graphs in small spaces in this paper we discuss how such succinct graph descriptions can be used to speed up the solution of graph problems we present the bottom up method that solves graph problems without expanding the hierarchical description this allows solutions that are efficient in terms of the hierarchical graph description instead of the size of the expanded graph we exemplify the method by giving efficient solutions to connectivity problems on hierarchical graphs our results have applications in computer aided design for integrated circuit design and other engineering problems

nonanticipative duality relaxations and formulations for chance constrained stochastic programs we propose two new lagrangian dual problems for chance constrained stochastic programs based on relaxing nonanticipativity constraints we compare the strength of the proposed dual bounds and demonstrate that they are superior to the bound obtained from the continuous relaxation of a standard mixed integer programming mip formulation for a given dual solution the associated lagrangian relaxation bounds can be calculated by solving a set of single scenario subproblems and then solving a single knapsack problem we also derive two new primal mip formulations and demonstrate that for chance constrained linear programs the continuous relaxations of these formulations yield bounds equal to the proposed dual bounds we propose a new heuristic method and two new exact algorithms based on these duals and formulations the first exact algorithm applies to chance constrained binary programs and uses either of the proposed dual bounds in concert with cuts that eliminate solutions found by the subproblems the second exact method is a branch and cut algorithm for solving either of the primal formulations our computational results indicate that the proposed dual bounds and heuristic solutions can be obtained efficiently and the gaps between the best dual bounds and the heuristic solutions are small

convex programming for disjunctive convex optimization given a finite number of closed convex sets whose algebraic representation is known we study the problem of finding the minimum of a convex function on the closure of the convex hull of the union of those sets we derive an algebraic characterization of the feasible region in a higher dimensional space and propose a solution procedure akin to the interior point approach for convex programming

derivatives of probability functions and integrals over sets given by inequalities abstract probability functions depending upon their parameters are reduced to integrals calculated over sets given by many inequalities a new general formula for the differentiation of such integrals is proposed a gradient of the integral is represented as the sum of integrals taken over a volume and over a surface these results are used to calculate the sensitivities of probability functions and also for chance constrained optimization

a partial k arboretum of graphs with bounded treewidth the notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies this survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class also some mutual relations between such classes are discussed

mip reformulations of the probabilistic set covering problem in this paper we address the following probabilistic version psc of the set covering problem min cx mathbb p ax ge xi ge p x in 0 1 n where a is a 0 1 matrix xi is a random 0 1 vector and p in 0 1 is the threshold probability level we introduce the concepts of p inefficiency and polarity cuts while the former is aimed at deriving an equivalent mip reformulation of psc the latter is used as a strengthening device to obtain a stronger formulation simplifications of the mip model which result when one of the following conditions hold are briefly discussed a is a balanced matrix a has the circular ones property the components of xi are pairwise independent the distribution function of xi is a stationary distribution or has the disjunctive shattering property we corroborate our theoretical findings by an extensive computational experiment on a test bed consisting of almost 10 000 probabilistic instances this test bed was created using deterministic instances from the literature and consists of probabilistic variants of the set covering model and capacitated versions of facility location warehouse location and k median models our computational results show that our procedure is orders of magnitude faster than any of the existing approaches to solve psc and in many cases can reduce hours of computing time to a fraction of a second

maximizing a class of utility functions over the vertices of a polytope given a polytope x a monotone concave univariate function g and two vectors c and d we study the discrete optimization problem of finding a vertex of x that maximizes the utility function c x g d x this problem has numerous applications in combinatorial optimization with a probabilistic objective including estimation of project duration with stochastic times in reliability models in multinomial logit models and in robust optimization we show that the problem is x1d4a9 x1d4ab hard for any strictly concave function g even for simple polytopes such as the uniform matroid assignment and path polytopes and propose a 1 2 approximation algorithm for it we discuss improvements for special cases where g is the square root log utility negative exponential utility and multinomial logit probability function in particular for the square root function the approximation ratio is 4 5 we also propose a 1 25 approximation algorithm for a class of minimization problems in which the maximization of the utility func

decision algorithms for multiplayer noncooperative games of incomplete information abstract extending the complexity results of reif 1 2 for two player games of incomplete information this paper see also 3 presents algorithms for deciding the outcome for various classes of multiplayer games of incomplete information i e deciding whether or not a team has a winning strategy for a particular game our companion paper 4 shows that these algorithms are indeed asymptotically optimal by providing matching lower bounds the classes of games to which our algorithms are applicable include games which were not previously known to be decidable we apply our algorithms to provide alternative upper bounds and new time space trade offs on the complexity of multiperson alternating turing machines 3 we analyze the algorithms to characterize the space complexity of multiplayer games in terms of the complexity of deterministic computation on turing machines in hierarchical multiplayer games each additional clique subset of players with the same information increases the complexity of the outcome problem by a further exponential we show that an s n space bounded k player game of incomplete information has a deterministic time upper bound of k 1 repeated exponentials of s n furthermore s n space bounded k player blindfold games have a deterministic space upper bound of k repeated exponentials of s n this paper proves that this exponential blow up can occur we also show that time bounded games do not exhibit such hierarchy a t n time bounded blindfold multiplayer game as well as a t n time bounded multiplayer game of incomplete information has a deterministic space bound of t n

price of correlations in stochastic optimization when decisions are made in the presence of high dimensional stochastic data handling joint distribution of correlated random variables can present a formidable task both in terms of sampling and estimation as well as algorithmic complexity a common heuristic is to estimate only marginal distributions and substitute joint distribution by independent product distribution in this paper we study possible loss incurred on ignoring correlations through a distributionally robust stochastic programming model and we quantify that loss as price of correlations poc using techniques of cost sharing from game theory we identify a wide class of problems for which poc has a small upper bound to our interest this class will include many stochastic convex programs uncapacitated facility location steiner tree and submodular functions suggesting that the intuitive approach of assuming independent distribution acceptably approximates the robust model for these stochastic optimization problems additionally we demonstrate hardness of bounding poc via examples of subadditive and supermodular functions that have large poc we find that our results are also useful for solving many deterministic optimization problems like welfare maximization k dimensional matching and transportation problems under certain conditions

a chance constrained programming framework to handle uncertainties in radiation therapy treatment planning abstract a stochastic programming framework is proposed for radiation therapy treatment planning this framework takes into account uncertainty in setting up or positioning patients identically day to day during the treatment period because uncertainties are unavoidable constraint violations are tolerated to some degree in practice under this assumption a chance constrained programming ccp framework is developed to handle setup uncertainties in treatment planning the proposed framework can be employed under different distributional assumptions the goal of the proposed approach is to maximize both the statistical confidence level of a treatment plan and the homogeneity of the dose distributions this novel perspective provides a user centric and personalized optimization model that allows a trade off between sufficient tumor coverage and sparing healthy tissues under uncertainty we describe testing of the performance of the proposed ccp models in terms of plan quality robustness and homogeneity and confidence level of the constraints using clinical data for a prostate cancer patient optimized ccp plans are also compared to plans developed using a deterministic approach that does not take uncertainties into account numerical experiments confirmed that the ccp is able to control setup uncertainties in target coverage and sparing of organs at risk

an integer optimization approach to a probabilistic reserve site selection problem interest in protecting natural areas is increasing as development pressures and conflicting land uses threaten and fragment ecosystems a variety of quantitative approaches have been developed to help managers select sites for biodiversity protection the problem is often formulated to select the set of reserve sites that maximizes the number of species or ecological communities that are represented subject to an upper bound on the number or area of selected sites most formulations assume that information about the presence or absence of species in the candidate sites is known with certainty because complete information typically is lacking we developed a reserve selection formulation that incorporates probabilistic presence absence data the formulation was a discrete 0 1 optimization model that maximized the number of represented vegetation communities subject to a budget constraint where a community was considered represented if its probability of occurrence in the set of selected sites exceeded a specified minimum reliability threshold although the formulation was nonlinear a log transformation allowed us to represent the problem in a linear format that could be solved using exact optimization methods the formulation was tested using a moderately sized reserve selection problem based on data from the superior national forest in minnesota

stochastic model of tumor induced angiogenesis ensemble averages and deterministic equations a recent conceptual model of tumor driven angiogenesis including branching elongation and anastomosis of blood vessels captures some of the intrinsic multiscale structures of this complex system yet allowing one to extract a deterministic integro partial differential description of the vessel tip density phys rev e 90 062716 2014 here we solve the stochastic model show that ensemble averages over many realizations correspond to the deterministic equations and fit the anastomosis rate coefficient so that the total number of vessel tips evolves similarly in the deterministic and ensemble averaged stochastic descriptions

games on interval and permutation graph representations we describe combinatorial games on graphs in which two players antagonistically build a representation of a subgraph of a given graph we show that for a large class of these games determining whether a given instance is a winning position for the next player is pspace hard in contrast we give polynomial time algorithms for solving some versions of the games on trees

maximizing a class of submodular utility functions given a finite ground set n and a value vector a in mathbb r n we consider optimization problems involving maximization of a submodular set utility function of the form h s f left sum i in s a i right s subseteq n where f is a strictly concave increasing differentiable function this utility function appears frequently in combinatorial optimization problems when modeling risk aversion and decreasing marginal preferences for instance in risk averse capital budgeting under uncertainty competitive facility location and combinatorial auctions these problems can be formulated as linear mixed 0 1 programs however the standard formulation of these problems using submodular inequalities is ineffective for their solution except for very small instances in this paper we perform a polyhedral analysis of a relevant mixed integer set and by exploiting the structure of the utility function h strengthen the standard submodular formulation significantly we show the lifting problem of the submodular inequalities to be a submodular maximization problem with a special structure solvable by a greedy algorithm which leads to an easily computable strengthening by subadditive lifting of the inequalities computational experiments on expected utility maximization in capital budgeting show the effectiveness of the new formulation