dc programming and dca thirty years of developments the year 2015 marks the 30th birthday of dc difference of convex functions programming and dca dc algorithms which constitute the backbone of nonconvex programming and global optimization in this article we offer a short survey on thirty years of developments of these theoretical and algorithmic tools the survey is comprised of three parts in the first part we present a brief history of the field while in the second we summarize the state of the art results and recent advances we focus on main theoretical results and dca solvers for important classes of difficult nonconvex optimization problems and then give an overview of real world applications whose solution methods are based on dca the third part is devoted to new trends and important open issues as well as suggestions for future developments

analysis and design of impulsive control systems some sufficient conditions for asymptotic stability of impulsive control systems with impulses at fixed times have recently been presented in this paper we derive some less conservative conditions for asymptotic stability of such impulsive control systems and the results are used to design impulsive control for a class of nonlinear systems the class of nonlinear systems considered is also extended

convergence of time stepping schemes for passive and extended linear complementarity systems generalizing recent results in m k camlibel complementarity methods in the analysis of piecewise linear dynamical systems ph d thesis center for economic research tilburg university tilburg the netherlands 2001 m k camlibel w p m h heemels and j m schumacher ieee trans circuits systems i fund theory appl 49 2002 pp 349 357 and j s pang and d stewart math program ser a 113 2008 pp 345 424 this paper provides an in depth analysis of time stepping methods for solving initial value and boundary value non lipschitz linear complementarity systems lcss under passivity and broader assumptions the novelty of the methods and their analysis lies in the use of least norm solutions in the discrete time linear complementarity subproblems arising from the numerical scheme these subproblems are not necessarily monotone and are not guaranteed to have convex solution sets among the principal results it is shown that using such least norm solutions of the discrete time subproblems an implicit euler scheme is convergent for passive initial value lcss generalizations under a strict copositivity assumption and for boundary value lcss are also established

delay dependent absolute stability of lurie control systems with multiple time delays the stability of lurie control systems with multiple time delays is investigated the system with multiple time delays is transformed then a suitable lyapunov function is selected the delay dependent absolutely stable condition for lurie control systems with multiple time delays is obtained by applying the technique of analyzing inequality and the method of decomposing the matrices the condition is based on the linear matrix inequality finally a numerical example is given to demonstrate the derived condition is less conservative than that given in the literature

on the pervasiveness of difference convexity in optimization and statistics with the increasing interest in applying the methodology of difference of convex dc optimization to diverse problems in engineering and statistics this paper establishes the dc property of many functions in various areas of applications not previously known to be of this class motivated by a quadratic programming based recourse function in two stage stochastic programming we show that the optimal value function of a copositive thus not necessarily convex quadratic program is dc on the domain of finiteness of the program when the matrix in the objective function s quadratic term and the constraint matrix are fixed the proof of this result is based on a dc decomposition of a piecewise hbox lc 1 function i e functions with lipschitz gradients armed with these new results and known properties of dc functions existed in the literature we show that many composite statistical functions in risk analysis including the value at risk var conditional value at risk cvar optimized certainty equivalent and the expectation based var based and cvar based random deviation functionals are all dc adding the known class of dc surrogate sparsity functions that are employed as approximations of the ell 0 function in statistical learning our work significantly expands the classes of dc functions and positions them for fruitful applications

impulsive control in this paper the concept and principles of impulsive control are presented impulsive control can be applied to a class of systems whose state variables are changeable in a very short time period in particular the asymptotic stability of the impulsive control of a nonlinear system is given

impulsive control of lurie systems in this paper the impulsive control scheme of lurie systems is represented using lyapunov s direct method and comparison technique we present novel conditions under which impulsively controlled lurie systems are asymptotically stable and give the estimate of the upper bound of impulse interval in particular globally exponential stability of impulsive control of lurie systems is presented impulsive controller design with examples is provided to show the feasibility and effectiveness of our method finally

an lpcc approach to nonconvex quadratic programs filling a gap in nonconvex quadratic programming this paper shows that the global resolution of a feasible quadratic program qp which is not known a priori to be bounded or unbounded below can be accomplished in finite time by solving two linear programs with linear complementarity constraints i e lpccs specifically this task can be divided into two lpccs the first confirms whether the qp is bounded below on the feasible set and if not computes a feasible ray on which the qp is unbounded the second lpcc computes a globally optimal solution if it exists by identifying a stationary point that yields the best quadratic objective value in turn the global resolution of these lpccs can be accomplished by a parameter free mixed integer programming based finitely terminating algorithm developed recently by the authors which can be enhanced in this context by a new kind of valid cut derived from the second order conditions of the qp and by exploiting the special structure of the lpccs throughout our treatment makes no boundedness assumption of the qp this is a significant departure from much of the existing literature which consistently employs the boundedness of the feasible set as a blanket assumption the general theory is illustrated by 3 classes of indefinite problems qps with simple upper and lower bounds existence of optimal solutions is guaranteed same qps with an additional inequality constraint extending the case of simple bound constraints and nonnegatively constrained copositive qps no guarantee of the existence of an optimal solution we also present numerical results to support the special cuts obtained due to the qp connection

a new regularization method for mathematical programs with complementarity constraints with strong convergence properties mathematical programs with equilibrium or complementarity constraints mpecs form a difficult class of optimization problems the feasible set has a very special structure and violates most of the standard constraint qualifications therefore one typically applies specialized algorithms in order to solve mpecs one very prominent class of specialized algorithms are the regularization or relaxation methods the first regularization method for mpecs is due to scholtes siam j optim 11 2001 pp 918 936 but in the meantime there exist a number of different regularization schemes which try to relax the difficult constraints in different ways however almost all regularization methods converge to c stationary points only which is a very weak stationarity concept an exception is a recent method by kadrani dussault and benchakroun siam j optim 20 2009 pp 78 103 whose limit points are shown to be m stationary here we provide a new regularization method which also converges to m sta

error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem consider the affine variational inequality problem it is shown that the distance to the solution set from a feasible point near the solution set can be bounded by the norm of a natural residual at that point this bound is then used to prove linear convergence of a matrix splitting algorithm for solving the symmetric case of the problem this latter result improves upon a recent result of luo and tseng that further assumes the problem to be monotone

computing b stationary points of nonsmooth dc programs motivated by a class of applied problems arising from physical layer based security in a digital communication system in particular by a secrecy sum rate maximization problem this paper studies a nonsmooth difference of convex dc minimization problem the contributions of this paper are i clarify several kinds of stationary solutions and their relations ii develop and establish the convergence of a novel algorithm for computing a d stationary solution of a problem with a convex feasible set that is arguably the sharpest kind among the various stationary solutions iii extend the algorithm in several directions including a randomized choice of the subproblems that could help the practical convergence of the algorithm a distributed penalty approach for problems whose objective functions are sums of dc functions and problems with a specially structured nonconvex dc constraint for the latter class of problems a pointwise slater constraint qualification is introduced that facilitates the ver

convex relaxations of non convex mixed integer quadratically constrained programs extended formulations this paper addresses the problem of generating strong convex relaxations of mixed integer quadratically constrained programming miqcp problems miqcp problems are very difficult because they combine two kinds of non convexities integer variables and non convex quadratic constraints to produce strong relaxations of miqcp problems we use techniques from disjunctive programming and the lift and project methodology in particular we propose new methods for generating valid inequalities from the equation y x x t we use the non convex constraint y x x t preccurlyeq 0 to derive disjunctions of two types the first ones are directly derived from the eigenvectors of the matrix y x x t with positive eigenvalues the second type of disjunctions are obtained by combining several eigenvectors in order to minimize the width of the disjunction we also use the convex sdp constraint y x x t succcurlyeq 0 to derive convex quadratic cuts and we combine both approaches in a cutting plane algorithm we present computational results to illustrate our findings

sufficient and necessary conditions for absolute stability of time delayed lurie control systems in this paper we present some sufficient and necessary conditions for the absolute stability of time delayed lurie control systems which improve the results we obtained before for the absolute stability of direct control indirect control and critical control of lurie systems we also derived certain simple and easy applicable algebraic sufficient conditions for the absolute stability of time delayed lurie control systems which provide a convenient tool for practical control engineers in designing absolutely stable systems or stabilizing nonlinear control systems

an outer approximation algorithm for a class of mixed integer nonlinear programs an outer approximation algorithm is presented for solving mixed integer nonlinear programming problems of a particular class linearity of the integer or discrete variables and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure based on principles of decomposition outer approximation and relaxation the proposed algorithm effectively exploits the structure of the problems and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a mixed integer linear master program convergence and optimality properties of the algorithm are presented as well as a general discussion on its implementation numerical results are reported for several example problems to illustrate the potential of the proposed algorithm for programs in the class addressed in this paper finally a theoretical comparison with generalized benders decomposition is presented on the lower bounds predicted by the relaxed master programs

a symmetrical linear maxmin approach to disjoint bilinear programming the disjoint bilinear programming problem can be reformulated using two distinct linear maxmin programming problems there is a simple bijection between the optimal solutions of the reformulations and the bilinear problem moreover the number of local optima of the reformulations is less or equal to that of the bilinear problem in that sense the reformulations do not introduce additional complexity necessary optimality conditions complementarity and monotonicity of both reformulations are used to obtain a finitely convergent and exact branch and bound algorithm for the bilinear problem determining if the optimal value of a bilinear programming problem is bounded is shown to be strongly np complete the proposed algorithm may be used to answer this question

new analysis on sparse solutions to random standard quadratic optimization problems and extensions the standard quadratic optimization problem stqp refers to the problem of minimizing a quadratic form over the standard simplex such a problem arises from numerous applications and is known to be np hard in a recent paper chen x peng j zhang s 2013 sparse solutions to random standard quadratic optimization problems math programming 141 1 2 273 293 chen et al showed that with a high probability close to 1 stqps with random data have sparse optimal solutions when the associated data matrix is randomly generated from a certain distribution such as uniform and exponential distributions in this paper we present a new analysis for random stqps combining probability inequalities derived from the first order and second order optimality conditions the new analysis allows us to significantly improve the probability bounds more important it allows us to handle normal distributions which is left open in chen et al 2013 the existence of sparse approximate solutions to convex stqps and extens

mathematical programs with complementarity constraints we study mathematical programs with complementarity constraints several stationarity concepts based on a piecewise smooth formulation are presented and compared the concepts are related to stat

on conic qpccs conic qcqps and completely positive programs this paper studies several classes of nonconvex optimization problems defined over convex cones establishing connections between them and demonstrating that they can be equivalently formulated as convex completely positive programs the problems being studied include a conic quadratically constrained quadratic program qcqp a conic quadratic program with complementarity constraints qpcc and rank constrained semidefinite programs our results do not make any boundedness assumptions on the feasible regions of the various problems considered the first stage in the reformulation is to cast the problem as a conic qcqp with just one nonconvex constraint q x le 0 q x 0 where q x is nonnegative over the convex conic and linear constraints so the problem fails the slater constraint qualification a conic qpcc has such a structure we prove the converse namely that any conic qcqp satisfying a constraint qualification can be expressed as an equivalent conic qpcc the second stage of the reformulation lifts the problem to a completely positive program and exploits and generalizes a result of burer we also show that a frank wolfe type result holds for a subclass of this class of conic qcqps further we derive necessary and sufficient optimality conditions for nonlinear programs where the only nonconvex constraint is a quadratic constraint of the structure considered elsewhere in the paper

technical communique a note on connectivity of multi agent systems with proximity graphs and linear feedback protocol in this note we investigate the possibility of simplifying the connectivity verification for multi agent systems with proximity graphs and the linear feedback protocol a lyapunov like approach is developed for analyzing the monotonicity of the largest edge length in the network based on this approach we provide an initial connectivity based sufficient condition for dynamic connectivity in particular when each initial node degree is not less than three fourth of the graph order initial network connectivity implies dynamic connectivity

logic based benders decomposition benders decomposition uses a strategy of learning from one s mistakes the aim of this paper is to extend this strategy to a much larger class of problems the key is to generalize the linear programming dual used in the classical method to an inference dual solution of the inference dual takes the form of a logical deduction that yields benders cuts the dual is therefore very different from other generalized duals that have been proposed the approach is illustrated by working out the details for propositional satisfiability and 0 1 programming problems computational tests are carried out for the latter but the most promising contribution of logic based benders may be to provide a framework for combining optimization and constraint programming methods

stationarity conditions and constraint qualifications for mathematical programs with switching constraints in optimal control switching structures demanding at most one control to be active at any time instance appear frequently discretizing such problems a so called mathematical program with switching constraints is obtained although these problems are related to other types of disjunctive programs like optimization problems with complementarity or vanishing constraints their inherent structure makes a separate consideration necessary since standard constraint qualifications are likely to fail at the feasible points of switching constrained optimization problems stationarity notions which are weaker than the associated karush kuhn tucker conditions need to be investigated in order to find applicable necessary optimality conditions furthermore appropriately tailored constraint qualifications need to be formulated in this paper we introduce suitable notions of weak mordukhovich and strong stationarity for mathematical programs with switching constraints and present some associated constraint qualifications our findings are exploited to state necessary optimality conditions for discretized optimal control problems with switching constraints furthermore we apply our results to optimization problems with either or constraints first a novel reformulation of such problems using switching constraints is presented second the derived surrogate problem is exploited to obtain necessary optimality conditions for the original program

on the global convergence of a filter sqp algorithm a mechanism for proving global convergence in sqp filter methods for nonlinear programming nlp is described such methods are characterized by their use of the dominance concept of multiobjective optimization instead of a penalty parameter whose adjustment can be problematic the main point of interest is to demonstrate how convergence for nlp can be induced without forcing sufficient descent in a penalty type merit function the proof relates to a prototypical algorithm within which is allowed a range of specific algorithm choices associated with the hessian matrix representation updating the trust region radius and feasibility restoration

on the rendezvous problem for multiple nonholonomic agents in this note a decentralized feedback control strategy that drives a system of multiple nonholonomic unicycles to a rendezvous point in terms of both position and orientation is introduced the proposed nonholonomic control law is discontinuous and time invariant and using tools from nonsmooth lyapunov theory and graph theory the stability of the overall system is examined similarly to the linear case the convergence of the multi agent system relies on the connectivity of the communication graph that represents the inter agent communication topology the control law is first defined in order to guarantee connectivity maintenance for an initially connected communication graph moreover the cases of static and dynamic communication topologies are treated as corollaries of the proposed framework

brief paper a novel approach to coordination of multiple robots with communication failures via proximity graph in this paper we investigate the coordination problem of multiple robots with limited communication ranges and communication failures a novel rendezvous algorithm via proximity graph is developed so that the robot group can achieve rendezvous when the communication links satisfy an ergodic assumption the convergence proof of the algorithm is established based on the tools from rooted graph theory the effectiveness of the algorithm is illustrated by numerical examples in a 3d space

mathematical programs with cardinality constraints reformulation by complementarity type conditions and a regularization method optimization problems with cardinality constraints are very difficult mathematical programs which are typically solved by global techniques from discrete optimization here we introduce a mixed integer formulation whose standard relaxation still has the same solutions in the sense of global minima as the underlying cardinality constrained problem the relation between the local minima is also discussed in detail since our reformulation is a minimization problem in continuous variables it allows us to apply ideas from that field to cardinality constrained problems here in particular we therefore also derive suitable stationarity conditions and suggest an appropriate regularization method for the solution of optimization problems with cardinality constraints this regularization method is shown to be globally convergent to a mordukhovich stationary point extensive numerical results are given to illustrate the behavior of this method

on synchronous robotic networks part ii time complexity of rendezvous and deployment algorithms this paper analyzes a number of basic coordination algorithms running on synchronous robotic networks we provide upper and lower bounds on the time complexity of the move toward average and circumcenter laws both achieving rendezvous and of the centroid law achieving deployment over a region of interest the results are derived via novel analysis methods including a set of results on the convergence rates of linear dynamical systems defined by tridiagonal toeplitz and circulant matrices

a study of the difference of convex approach for solving linear programs with complementarity constraints this paper studies the difference of convex dc penalty formulations and the associated difference of convex algorithm dca for computing stationary solutions of linear programs with complementarity constraints lpccs we focus on three such formulations and establish connections between their stationary solutions and those of the lpcc improvements of the dca are proposed to remedy some drawbacks in a straightforward adaptation of the dca to these formulations extensive numerical results including comparisons with an existing nonlinear programming solver and the mixed integer formulation are presented to elucidate the effectiveness of the overall dc approach

a finite branch and bound algorithm for nonconvex quadratic programming via semidefinite relaxations existing global optimization techniques for nonconvex quadratic programming qp branch by recursively partitioning the convex feasible set and thus generate an infinite number of branch and bound nodes an open question of theoretical interest is how to develop a finite branch and bound algorithm for nonconvex qp one idea which guarantees a finite number of branching decisions is to enforce the first order karush kuhn tucker kkt conditions through branching in addition such an approach naturally yields linear programming lp relaxations at each node however the lp relaxations are unbounded a fact that precludes their use in this paper we propose and study semidefinite programming relaxations which are bounded and hence suitable for use with finite kkt branching computational results demonstrate the practical effectiveness of the method with a particular highlight being that only a small number of nodes are required

a sequential algorithm for solving nonlinear optimization problems with chance constraints an algorithm is presented for solving nonlinear optimization problems with chance constraints i e those in which a constraint involving an uncertain parameter must be satisfied with at least a minimum probability in particular the algorithm is designed to solve cardinality constrained nonlinear optimization problems that arise in sample average approximations of chance constrained problems as well as in other applications in which it is only desired to enforce a minimum number of constraints the algorithm employs a novel exact penalty function which is minimized sequentially by solving quadratic optimization subproblems with linear cardinality constraints properties of minimizers of the penalty function in relation to minimizers of the corresponding nonlinear optimization problem are presented and convergence of the proposed algorithm to stationarity with respect to the penalty function is proved the effectiveness of the algorithm is demonstrated through numerical experiments with a nonlinear ca

computational study of a family of mixed integer quadratic programming problems we present computational experience with a branch and cut algorithm to solve quadratic programming problems where there is an upper bound on the number of positive variables such problems arise in financial applications the algorithm solves the largest real life problems in a few minutes of run time

a branch and cut algorithm for nonconvex quadratic programs with box constraints we present the implementation of a branch and cut algorithm for bound constrained nonconvex quadratic programs we use a class of inequalities developed in 12 as cutting planes we present various branching strategies and compare the algorithm to several other methods to demonstrate its effectiveness

combinatorial benders cuts for mixed integer linear programming mixed integer programs mips involving logical implications modeled through big m coefficients are notoriously among the hardest to solve in this paper we propose and analyze computationally an automatic problem reformulation of quite general applicability aimed at removing the model dependency on the big m coefficients our solution scheme defines a master integer linear problem ilp with no continuous variables which contains combinatorial information on the feasible integer variable combinations that can be distilled from the original mip model the master solutions are sent to a slave linear program lp which validates them and possibly returns combinatorial inequalities to be added to the current master ilp the inequalities are associated to minimal or irreducible infeasible subsystems of a certain linear system and can be separated efficiently in case the master solution is integer the overall solution mechanism closely resembles the benders one but the cuts we produce are purely co

structural properties of affine sparsity constraints we introduce a new constraint system for sparse variable selection in statistical learning such a system arises when there are logical conditions on the sparsity of certain unknown model parameters that need to be incorporated into their selection process formally extending a cardinality constraint an affine sparsity constraint asc is defined by a linear inequality with two sets of variables one set of continuous variables and the other set represented by their nonzero patterns this paper aims to study an asc system consisting of finitely many affine sparsity constraints we investigate a number of fundamental structural properties of the solution set of such a non standard system of inequalities including its closedness and the description of its closure continuous approximations and their set convergence and characterizations of its tangent cones for use in optimization based on the obtained structural properties of an asc system we investigate the convergence of b ouligand stationary solutions when the asc is approximated by surrogates of the step ell 0 function commonly employed in sparsity representation our study lays a solid mathematical foundation for solving optimization problems involving these affine sparsity constraints through their continuous approximations

consensus seeking in multiagent systems under dynamically changing interaction topologies this note considers the problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies both discrete and continuous update schemes are proposed for information consensus this note shows that information consensus under dynamically changing interaction topologies can be achieved asymptotically if the union of the directed interaction graphs have a spanning tree frequently enough as the system evolves

absolute stability of lurie systems with impulsive effects abstract this paper studies absolute stability of lurie systems with impulsive effects using the method of lyapunov functions and the variation of parameters technique we establish sufficient conditions for absolute stability

impulsive control of lorenz system abstract in this paper an impulsive control scheme of the lorenz system is presented first we use the theory of impulsive differential equation to find conditions under which impulsively controlled lorenz system is asymptotically stable then we give the estimate of the upper bound of impulse interval for asymptotically stable control numerical experimental results are presented

lyapunov based approach to multiagent systems with switching jointly connected interconnection this note addresses a coordination problem of a multiagent system with jointly connected interconnection topologies neighbor based rules are adopted to realize local control strategies for these continuous time autonomous agents described by double integrators although the interagent connection structures vary over time and related graphs may not be connected a sufficient condition to make all the agents converge to a common value is given for the problem by a proposed lyapunov based approach and related space decomposition technique

a polyhedral study of nonconvex quadratic programs with box constraints by reformulating quadratic programs using necessary optimality conditions we obtain a linear program with complementarity constraints for the case where the only constraints are bounds on the variables we study a relaxation based on a subset of the optimality conditions by characterizing its convex hull we obtain a large class of valid inequalities these inequalities are used in a branch and cut scheme see 13