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In this paper, we consider the problem of computing the rank of a block-structured symbolic matrix (a generic partitioned matrix) A = (A α β x α β) , where A α β is a 2 × 2 matrix over a field F and x α β is an indeterminate for α = 1 , 2 , ⋯ , μ and β = 1 , 2 , ⋯ , ν . This problem can be viewed as an algebraic generalization of the bipartite matching problem and was considered by Iwata and Murota (SIAM J Matrix Anal Appl 16(3):719–734, 1995). Recent interests in this problem lie in the connection with non-commutative Edmonds' problem by Ivanyos et al. (Comput Complex 27:561–593, 2018) and Garg et al. (Found. Comput. Math. 20:223–290, 2020), where a result by Iwata and Murota implicitly states that the rank and non-commutative rank (nc-rank) are the same for this class of symbolic matrices. The main result of this paper is a simple and combinatorial O ((μ ν) 2 min { μ , ν }) -time algorithm for computing the symbolic rank of a (2 × 2) -type generic partitioned matrix of size 2 μ × 2 ν . Our algorithm is inspired by the Wong sequence algorithm by Ivanyos et al. for the nc-rank of a general symbolic matrix, and requires no blow-up operation, no field extension, and no additional care for bounding the bit-size. Moreover it naturally provides a maximum rank completion of A for an arbitrary field F . [ABSTRACT FROM AUTHOR]

In this paper we consider a class of structured nonsmooth difference-of-convex (DC) minimization in which the first convex component is the sum of a smooth and nonsmooth functions while the second convex component is the supremum of possibly infinitely many convex smooth functions. We first propose an inexact enhanced DC algorithm for solving this problem in which the second convex component is the supremum of finitely many convex smooth functions, and show that every accumulation point of the generated sequence is an (α , η) -D-stationary point of the problem, which is generally stronger than an ordinary D-stationary point. In addition, inspired by the recent work (Pang et al. in Math Oper Res 42(1):95–118, 2017; Wen et al. in Comput Optim Appl 69(2):297–324, 2018), we propose two proximal DC algorithms with extrapolation for solving this problem. We show that every accumulation point of the solution sequence generated by them is an (α , η) -D-stationary point of the problem, and establish the convergence of the entire sequence under some suitable assumption. We also introduce a concept of approximate (α , η) -D-stationary point and derive iteration complexity of the proposed algorithms for finding an approximate (α , η) -D-stationary point. In contrast with the DC algorithm (Pang et al. 2017), our proximal DC algorithms have much simpler subproblems and also incorporate the extrapolation for possible acceleration. Moreover, one of our proximal DC algorithms is potentially applicable to the DC problem in which the second convex component is the supremum of infinitely many convex smooth functions. In addition, our algorithms have stronger convergence results than the proximal DC algorithm in Wen et al. (2018). [ABSTRACT FROM AUTHOR]

We analyze the convergence rate of the alternating direction method of multipliers (ADMM) for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints. Previous analysis of the ADMM typically assumes that the objective function is the sum of only two convex functions defined on two separable blocks of variables even though the algorithm works well in numerical experiments for three or more blocks. Moreover, there has been no rate of convergence analysis for the ADMM without strong convexity in the objective function. In this paper we establish the global R-linear convergence of the ADMM for minimizing the sum of any number of convex separable functions, assuming that a certain error bound condition holds true and the dual stepsize is sufficiently small. Such an error bound condition is satisfied for example when the feasible set is a compact polyhedron and the objective function consists of a smooth strictly convex function composed with a linear mapping, and a nonsmooth $$\ell _1$$ regularizer. This result implies the linear convergence of the ADMM for contemporary applications such as LASSO without assuming strong convexity of the objective function. [ABSTRACT FROM AUTHOR]

In this paper, we consider approximation algorithms for optimizing a generic multi-variate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, approximation theory, and portfolio selection. Since polynomial functions are non-convex, the problems under consideration are all NP-hard in general. In this paper we shall focus on polynomial-time approximation algorithms. In particular, we first study optimization of a multi-linear tensor function over the Cartesian product of spheres. We shall propose approximation algorithms for such problem and derive worst-case performance ratios, which are shown to be dependent only on the dimensions of the model. The methods are then extended to optimize a generic multi-variate homogeneous polynomial function with spherical constraint. Likewise, approximation algorithms are proposed with provable approximation performance ratios. Furthermore, the constraint set is relaxed to be an intersection of co-centered ellipsoids; namely, we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin, and propose polynomial-time approximation algorithms with provable worst-case performance ratios. Numerical results are reported, illustrating the effectiveness of the approximation algorithms studied. [ABSTRACT FROM AUTHOR]

Mixed-integer rounding (MIR) is a simple, yet powerful procedure for generating valid inequalities for mixed-integer programs. When used as cutting planes, MIR inequalities are very effective for mixed-integer programming problems with unbounded integer variables. For problems with bounded integer variables, however, cutting planes based on lifting techniques appear to be more effective. This is not surprising as lifting techniques make explicit use of the bounds on variables, whereas the MIR procedure does not. In this paper we describe a simple procedure, which we call mingling, for incorporating variable bound information into MIR. By explicitly using the variable bounds, the mingling procedure leads to strong inequalities for mixed-integer sets with bounded variables. We show that facets of mixed-integer knapsack sets derived earlier by superadditive lifting techniques can be obtained by the mingling procedure. In particular, the mingling inequalities developed in this paper subsume the continuous cover and reverse continuous cover inequalities of Marchand and Wolsey (Math Program 85:15–33, 1999) as well as the continuous integer knapsack cover and pack inequalities of Atamtürk (Math Program 98:145–175, 2003; Ann Oper Res 139:21–38, 2005). In addition, mingling inequalities give a generalization of the two-step MIR inequalities of Dash and Günlük (Math Program 105:29–53, 2006) under some conditions. [ABSTRACT FROM AUTHOR]

This paper presents a unified exact method for solving an extended model of the well-known Capacitated Vehicle Routing Problem (CVRP), called the Heterogenous Vehicle Routing Problem (HVRP), where a mixed fleet of vehicles having different capacities, routing and fixed costs is used to supply a set of customers. The HVRP model considered in this paper contains as special cases: the Single Depot CVRP, all variants of the HVRP presented in the literature, the Site-Dependent Vehicle Routing Problem (SDVRP) and the Multi-Depot Vehicle Routing Problem (MDVRP). This paper presents an exact algorithm for the HVRP based on the set partitioning formulation. The exact algorithm uses three types of bounding procedures based on the LP-relaxation and on the Lagrangean relaxation of the mathematical formulation. The bounding procedures allow to reduce the number of variables of the formulation so that the resulting problem can be solved by an integer linear programming solver. Extensive computational results over the main instances from the literature of the different variants of HVRPs, SDVRP and MDVRP show that the proposed lower bound is superior to the ones presented in the literature and that the exact algorithm can solve, for the first time ever, several test instances of all problem types considered. [ABSTRACT FROM AUTHOR]

This paper is concerned with characterizations of nonsmooth saddle critical points for numerical algorithm design. Most characterizations for nonsmooth saddle critical points in the literature focus on existence issue and are converted to solve global minimax problems. Thus they are not helpful for numerical algorithm design. Inspired by the results on computational theory and methods for finding multiple smooth saddle critical points in [14, 15, 19, 21, 23], a local minimax characterization for multiple nonsmooth saddle critical points in either a Hilbert space or a reflexive Banach space is established in this paper to provide a mathematical justification for numerical algorithm design. A local minimax algorithm for computing multiple nonsmooth saddle critical points is presented by its flow chart. [ABSTRACT FROM AUTHOR]

Let ( MQP) be a general mixed integer quadratic program that consists of minimizing a quadratic function subject to linear constraints. In this paper, we present a convex reformulation of ( MQP), i.e. we reformulate ( MQP) into an equivalent program, with a convex objective function. Such a reformulation can be solved by a standard solver that uses a branch and bound algorithm. We prove that our reformulation is the best one within a convex reformulation scheme, from the continuous relaxation point of view. This reformulation, that we call MIQCR (Mixed Integer Quadratic Convex Reformulation), is based on the solution of an SDP relaxation of ( MQP). Computational experiences are carried out with instances of ( MQP) including one equality constraint or one inequality constraint. The results show that most of the considered instances with up to 40 variables can be solved in 1 h of CPU time by a standard solver. [ABSTRACT FROM AUTHOR]

Developing a branching scheme that is compatible with the column generation procedure can be challenging. Application specific and generic schemes have been proposed in the literature, but they have their drawbacks. One generic scheme is to implement standard branching in the space of the compact formulation to which the Dantzig-Wolfe reformulation was applied. However, in the presence of multiple identical subsystems, the mapping to the original variable space typically induces symmetries. An alternative, in an application specific context, can be to expand the compact formulation to offer a wider choice of branching variables. Other existing generic schemes for use in branch-and-price imply modifications to the pricing problem. This is a concern because the pricing oracle on which the method relies might become obsolete beyond the root node. This paper presents a generic branching scheme in which the pricing oracle of the root node remains of use after branching (assuming that the pricing oracle can handle bounds on the subproblem variables). The scheme does not require the use of an extended formulation of the original problem. It proceeds by recursively partitioning the subproblem solution set. Branching constraints are enforced in the pricing problem instead of being dualized via Lagrangian relaxation, and the pricing problem is solved by a limited number of calls to the pricing oracle. This generic scheme builds on previously proposed approaches and unifies them. We illustrate its use on the cutting stock and bin packing problems. This is the first branch-and-price algorithm capable of solving such problems to integrality without modifying the subproblem or expanding its variable space. [ABSTRACT FROM AUTHOR]

An Adaptive Regularisation framework using Cubics (ARC) was proposed for unconstrained optimization and analysed in Cartis, Gould and Toint (Part I, Math Program, doi:, 2009), generalizing at the same time an unpublished method due to Griewank (Technical Report NA/12, 1981, DAMTP, University of Cambridge), an algorithm by Nesterov and Polyak (Math Program 108(1):177-205, 2006) and a proposal by Weiser, Deuflhard and Erdmann (Optim Methods Softw 22(3):413-431, 2007). In this companion paper, we further the analysis by providing worst-case global iteration complexity bounds for ARC and a second-order variant to achieve approximate first-order, and for the latter second-order, criticality of the iterates. In particular, the second-order ARC algorithm requires at most $${\mathcal{O}(\epsilon^{-3/2})}$$ iterations, or equivalently, function- and gradient-evaluations, to drive the norm of the gradient of the objective below the desired accuracy $${\epsilon}$$, and $${\mathcal{O}(\epsilon^{-3})}$$ iterations, to reach approximate nonnegative curvature in a subspace. The orders of these bounds match those proved for Algorithm 3.3 of Nesterov and Polyak which minimizes the cubic model globally on each iteration. Our approach is more general in that it allows the cubic model to be solved only approximately and may employ approximate Hessians. [ABSTRACT FROM AUTHOR]

MAX-2-SAT is one of the representative combinatorial problems and is known to be NP-hard. Given a set of m clauses on n propositional variables, where each clause contains at most two literals and is weighted by a positive real, MAX-2-SAT asks to find a truth assignment that maximizes the total weight of satisfied clauses. In this paper, we propose branch-and-bound exact algorithms for MAX-2-SAT utilizing three kinds of lower bounds. All lower bounds are based on a directed graph that represents conflicts among clauses, and two of them use a set covering representation of MAX-2-SAT. Computational comparisons on benchmark instances disclose that these algorithms are highly effective in reducing the number of search tree nodes as well as the computation time. [ABSTRACT FROM AUTHOR]

In this paper we consider the solution of certain convex integer minimization problems via greedy augmentation procedures. We show that a greedy augmentation procedure that employs only directions from certain Graver bases needs only polynomially many augmentation steps to solve the given problem. We extend these results to convex N-fold integer minimization problems and to convex 2-stage stochastic integer minimization problems. Finally, we present some applications of convex N-fold integer minimization problems for which our approach provides polynomial time solution algorithms. [ABSTRACT FROM AUTHOR]

In this paper we consider low-rank semidefinite programming (LRSDP) relaxations of combinatorial quadratic problems that are equivalent to the maxcut problem. Using the Gramian representation of a positive semidefinite matrix, the LRSDP problem can be formulated as the nonconvex nonlinear programming problem of minimizing a quadratic function with quadratic equality constraints. For the solution of this problem we propose a continuously differentiable exact merit function that exploits the special structure of the constraints and we use this function to define an efficient and globally convergent algorithm. Finally, we test our code on an extended set of instances of the maxcut problem and we report comparisons with other existing codes. [ABSTRACT FROM AUTHOR]

Celebrating the sixtieth anniversary since the zeroth International Symposium on Mathematical Programming was held in 1949, this paper discusses several promising paradigms in mathematical programming that have gained momentum in recent years but have yet to reach the main stream of the field. These are: competition, dynamics, and hierarchy. The discussion emphasizes the interplay between these paradigms and their connections with existing subfields including disjunctive, equilibrium, and nonlinear programming, and variational inequalities. We will describe the modeling approaches, mathematical formulations, and recent results of these paradigms, and sketch some open mathematical and computational challenges arising from the resulting optimization and equilibrium problems. Our goal is to elucidate the need for a systematic study of these problems and to inspire new research in the field. [ABSTRACT FROM AUTHOR]

Linear Programming, LP, problems with finite optimal value have a zero duality gap and a primal–dual strictly complementary optimal solution pair. On the other hand, there exist Semidefinite Programming, SDP, problems which have a nonzero duality gap (different primal and dual optimal values; not both infinite). The duality gap is assured to be zero if a constraint qualification, e.g., Slater’s condition (strict feasibility) holds. Measures of strict feasibility, also called distance to infeasibility, have been used in complexity analysis, and, it is known that (near) loss of strict feasibility results in numerical difficulties. In addition, there exist SDP problems which have a zero duality gap but no strict complementary primal–dual optimal solution. We refer to these problems as hard instances of SDP. The assumption of strict complementarity is essential for asymptotic superlinear and quadratic rate convergence proofs. In this paper, we introduce a procedure for generating hard instances of SDP with a specified complementarity nullity (the dimension of the common nullspace of primal–dual optimal pairs). We then show, empirically, that the complementarity nullity correlates well with the observed local convergence rate and the number of iterations required to obtain optimal solutions to a specified accuracy, i.e., we show, even when Slater’s condition holds, that the loss of strict complementarity results in numerical difficulties. We include two new measures of hardness that correlate well with the complementarity nullity. [ABSTRACT FROM AUTHOR]

Dynamic constraint aggregation is an iterative method that was recently introduced to speed up the linear relaxation solution process of set partitioning type problems. This speed up is mostly due to the use, at each iteration, of an aggregated problem defined by aggregating disjoint subsets of constraints from the set partitioning model. This aggregation is updated when needed to ensure the exactness of the overall approach. In this paper, we propose a new version of this method, called the multi-phase dynamic constraint aggregation method, which essentially adds to the original method a partial pricing strategy that involves multiple phases. This strategy helps keeping the size of the aggregated problem as small as possible, yielding a faster average computation time per iteration and fewer iterations. We also establish theoretical results that provide some insights explaining the success of the proposed method. Tests on the linear relaxation of simultaneous bus and driver scheduling problems involving up to 2,000 set partitioning constraints show that the partial pricing strategy speeds up the original method by an average factor of 4.5. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of two-stage stochastic risk management problems, which may be stated as follows. A decision-maker determines a set of binary first-stage decisions, after which a random event from a finite set of possible outcomes is realized. Depending on the realization of this outcome, a set of continuous second-stage decisions must then be made that attempt to minimize some risk function. We consider a hierarchy of multiple risk levels along with associated penalties for each possible scenario. The overall objective function thus depends on the cost of the first-stage decisions, plus the expected second-stage risk penalties. We develop a mixed-integer 0–1 programming model and adopt an automatic convexification procedure using the reformulation–linearization technique to recast the problem into a form that is amenable to applying Benders’ partitioning approach. As a principal computational expedient, we show how the reformulated higher-dimensional Benders’ subproblems can be efficiently solved via certain reduced-sized linear programs in the original variable space. In addition, we explore several key ingredients in our proposed procedure to enhance the tightness of the prescribed Benders’ cuts and the efficiency with which they are generated. Finally, we demonstrate the computational efficacy of our approaches on a set of realistic test problems. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish the second-order nonsmooth sufficient unconstrained optimality condition for so called ℓ-stable at some point functions and in this way to generalize some previous results in this direction. We provide the comparisons with other results by examples. [ABSTRACT FROM AUTHOR]

In this paper we present two theorems on the existence of a discrete zero point of a function from the n-dimensional integer lattice ℤ n to the n-dimensional Euclidean space ℝ n . The theorems differ in their boundary conditions. For both theorems we give a proof using a combinatorial lemma and present a constructive proof based on a simplicial algorithm that finds a discrete zero point within a finite number of steps. [ABSTRACT FROM AUTHOR]

A 0/±1 matrix is balanced if it does not contain a square submatrix with exactly two nonzero entries per row and per column in which the sum of all entries is 2 modulo 4. A 0/1 matrix is balanceable if its nonzero entries can be signed ±1 so that the resulting matrix is balanced. A signing algorithm due to Camion shows that the problems of recognizing balanced 0/±1 matrices and balanceable 0/1 matrices are equivalent. Conforti, Cornuéjols, Kapoor and Vušković gave an algorithm to test if a 0/±1 matrix is balanced. Truemper has characterized balanceable 0/1 matrices in terms of forbidden submatrices. In this paper we give an algorithm that explicitly finds one of these forbidden submatrices or shows that none exists. [ABSTRACT FROM AUTHOR]

The notion of weak sharp minima is an important tool in the analysis of the perturbation behavior of certain classes of optimization problems as well as in the convergence analysis of algorithms designed to solve these problems. It has been studied extensively by several authors. This paper is the second of a series on this subject where the basic results on weak sharp minima in Part I are applied to a number of important problems in convex programming. In Part II we study applications to the linear regularity and bounded linear regularity of a finite collection of convex sets as well as global error bounds in convex programming. We obtain both new results and reproduce several existing results from a fresh perspective. [ABSTRACT FROM AUTHOR]

In this paper we consider the NP-hard problem of finding a feasible solution (if any exists) for a generic MIP problem of the form min{ c T x: A x≥ b, x j integer ∀ j ∈ }. Trivially, a feasible solution can be defined as a point x* ∈ P:={ x: A x≥ b} that is equal to its rounding , where the rounded point is defined by := x* j if j ∈ and := x* j otherwise, and [·] represents scalar rounding to the nearest integer. Replacing “equal” with “as close as possible” relative to a suitable distance function Δ( x*, ), suggests the following Feasibility Pump (FP) heuristic for finding a feasible solution of a given MIP. We start from any x* ∈ P, and define its rounding . At each FP iteration we look for a point x* ∈ P that is as close as possible to the current by solving the problem min {Δ( x, ): x ∈ P}. Assuming Δ( x, ) is chosen appropriately, this is an easily solvable LP problem. If Δ( x*, )=0, then x* is a feasible MIP solution and we are done. Otherwise, we replace by the rounding of x*, and repeat. We report computational results on a set of 83 difficult 0-1 MIPs, using the commercial software ILOG-Cplex 8.1 as a benchmark. The outcome is that FP, in spite of its simple foundation, proves competitive with ILOG-Cplex both in terms of speed and quality of the first solution delivered. Interestingly, ILOG-Cplex could not find any feasible solution at the root node for 19 problems in our test-bed, whereas FP was unsuccessful in just 3 cases. [ABSTRACT FROM AUTHOR]

This paper examines Pareto optimality of solutions to multi-objective problems scalarized in the min-norm, compromise programming, generalized goal programming, or unrestricted min-max formulations. Issues addressed include, among others, uniqueness in solution or objective space, penalization for over-achievement of goals, min-max reformulation of goal programming, inferiority in Tchebycheff-norm minimization, strength and weakness of weighted-bound optimization, “quasi-satisficing” decision-making, just attaining or even over-passing the goals, trading off by modifying weights or goals, non-convex Pareto frontier. New general necessary and sufficient conditions for both Pareto optimality and weak Pareto optimality are presented. Various formulations are compared in theoretical performance with respect to the goal-point location. Ideas for advanced goal programming and interactive decision-making are introduced. [ABSTRACT FROM AUTHOR]

This paper presents some examples of ill-behaved central paths in convex optimization. Some contain infinitely many fixed length central segments; others manifest oscillations with infinite variation. These central paths can be encountered even for infinitely differentiable data. [ABSTRACT FROM AUTHOR]

The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. An instance of SAT is specified by a set of boolean variables and a propositional formula in conjunctive normal form. Given such an instance, the SAT problem asks whether there is a truth assignment to the variables such that the formula is satisfied. It is well known that SAT is in general NP-complete, although several important special cases can be solved in polynomial time. Semidefinite programming (SDP) refers to the class of optimization problems where a linear function of a matrix variableXis maximized (or minimized) subject to linear constraints on the elements ofXand the additional constraint thatXbe positive semidefinite. We are interested in the application of SDP to satisfiability problems, and in particular in how SDP can be used to detect unsatisfiability. In this paper we introduce a new SDP relaxation for the satisfiability problem. This SDP relaxation arises from the recently introduced paradigm of “higher liftings” for constructing semidefinite programming relaxations of discrete optimization problems. To derive the SDP relaxation, we first formulate SAT as an optimization problem involving matrices. Relaxing this formulation yields an SDP which significantly improves on the previous relaxations in the literature. The important characteristics of the SDP relaxation are its ability to prove that a given SAT formula is unsatisfiable independently of the lengths of the clauses in the formula, its potential to yield truth assignments satisfying the SAT instance if a feasible matrix of sufficiently low rank is computed, and the fact that it is more amenable to practical computation than previous SDPs arising from higher liftings. We present theoretical and computational results that support these claims. [ABSTRACT FROM AUTHOR]

The hub location problem with single assignment is the problem of locating hubs and assigning the terminal nodes to hubs in order to minimize the cost of hub installation and the cost of routing the traffic in the network. There may also be capacity restrictions on the amount of traffic that can transit by hubs. The aim of this paper is to investigate polyhedral properties of these problems and to develop a branch and cut algorithm based on these results. [ABSTRACT FROM AUTHOR]

Garg [10] gives two approximation algorithms for the minimum-cost tree spanning k vertices in an undirected graph. Recently Jain and Vazirani [15] discovered primal-dual approximation algorithms for the metric uncapacitated facility location and k-median problems. In this paper we show how Garg’s algorithms can be explained simply with ideas introduced by Jain and Vazirani, in particular via a Lagrangean relaxation technique together with the primal-dual method for approximation algorithms. We also derive a constant factor approximation algorithm for the k-Steiner tree problem using these ideas, and point out the common features of these problems that allow them to be solved with similar techniques. [ABSTRACT FROM AUTHOR]

This paper describes an active-set algorithm for large-scale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza [10]. The step computation is performed in two stages. In the first stage a linear program is solved to estimate the active set at the solution. The linear program is obtained by making a linear approximation to the ℓ1 penalty function inside a trust region. In the second stage, an equality constrained quadratic program (EQP) is solved involving only those constraints that are active at the solution of the linear program. The EQP incorporates a trust-region constraint and is solved (inexactly) by means of a projected conjugate gradient method. Numerical experiments are presented illustrating the performance of the algorithm on the CUTEr [1, 15] test set. [ABSTRACT FROM AUTHOR]

In this paper we present a primal-dual inexact infeasible interior-point algorithm for semidefinite programming problems (SDP). This algorithm allows the use of search directions that are calculated from the defining linear system with only moderate accuracy, and does not require feasibility to be maintained even if the initial iterate happened to be a feasible solution of the problem. Under a mild assumption on the inexactness, we show that the algorithm can find an ε-approximate solution of an SDP in O(n[sup 2]ln(1/ε)) iterations. This bound of our algorithm is the same as that of the exact infeasible interior point algorithms proposed by Y. Zhang. [ABSTRACT FROM AUTHOR]

Abstract. This paper describes components of a branch-and-cut algorithm for solving integer linear programs having a large symmetry group. It describes an isomorphism pruning algorithm and variable setting procedures using orbits of the symmetry group. Pruning and orbit computations are performed by backtracking procedures using a Schreier-Sims table for representing the symmetry group. Applications to hard set covering problems, generation of covering designs and error correcting codes are given. [ABSTRACT FROM AUTHOR]

In this paper a new class of proximal-like algorithms for solving monotone inclusions of the form T(x)∋0 is derived. It is obtained by applying linear multi-step methods (LMM) of numerical integration in order to solve the differential inclusion [ABSTRACT FROM AUTHOR]

The efficient combination of directions is a significant problem in line search methods that either use negative curvature, or wish to include additional information such as the gradient or different approximations to the Newton direction. In this paper we describe a new procedure to combine several of these directions within an interior-point primal-dual algorithm. Basically, we combine in an efficient manner a modified Newton direction with the gradient of a merit function and a direction of negative curvature, if it exists. We also show that the procedure is well-defined, and it has reasonable theoretical properties regarding the rate of convergence of the method. We also present numerical results from an implementation of the proposed algorithm on a set of small test problems from the CUTE collection. [ABSTRACT FROM AUTHOR]

In this paper we show that the so-called commutative class of primal-dual interior point algorithms which were designed by Monteiro and Zhang for semidefinite programming extends word-for-word to optimization problems over all symmetric cones. The machinery of Euclidean Jordan algebras is used to carry out this extension. Unlike some non-commutative algorithms such as the XS+SX method, this class of extensions does not use concepts outside of the Euclidean Jordan algebras. In particular no assumption is made about representability of the underlying Jordan algebra. As a special case, we prove polynomial iteration complexities for variants of the short-, semi-long-, and long-step path-following algorithms using the Nesterov-Todd, XS, or SX directions. [ABSTRACT FROM AUTHOR]

Abstract. This paper concerns with convergence properties of the classical proximal point algorithm for finding zeroes of maximal monotone operators in an infinite-dimensional Hilbert space. It is well known that the proximal point algorithm converges weakly to a solution under very mild assumptions. However, it was shown by Guler [11] that the iterates may fail to converge strongly in the infinite-dimensional case. We propose a new proximal-type algorithm which does converge strongly, provided the problem has a solution. Moreover, our algorithm solves proximal point subproblems inexactly, with a constructive stopping criterion introduced in [31]. Strong convergence is forced by combining proximal point iterations with simple projection steps onto intersection of two half spaces containing the solution set. Additional cost of this extra projection step is essentially negligible since it amounts, at most, to solving a linear system of two equations in two unknowns. [ABSTRACT FROM AUTHOR]

Presents information on the symposium "New Trends in Computational and Optimization Algorithms (NTOC2001)," organized and sponsored by the Institute of Statistical Mathematics, Tokyo, Japan from December 9 to 13, 2001. Topics of the symposium; Introduction to papers presented at the symposium.