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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]

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. [ABSTRACT FROM AUTHOR]

For 30 years after their invention half a century ago, cutting planes for integer programs have been an object of theoretical investigations that had no apparent practical use. When they finally proved their practical usefulness in the late eighties, that happened in the framework of branch and bound procedures, as an auxiliary tool meant to reduce the number of enumerated nodes. To this day, pure cutting plane methods alone have poor convergence properties and are typically not used in practice. Our reason for studying them is our belief that these negative properties can be understood and thus remedied only based on a thorough investigation of such procedures in their pure form. In this paper, the second in a sequence, we address some important issues arising when designing a computationally sound pure cutting plane method. We analyze the dual cutting plane procedure proposed by Gomory in 1958, which is the first (and most famous) convergent cutting plane method for integer linear programming. We focus on the enumerative nature of this method as evidenced by the relative computational success of its lexicographic version (as documented in our previous paper on the subject), and we propose new versions of Gomory's cutting plane procedure with an improved performance. In particular, the new versions are based on enumerative schemes that treat the objective function implicitly, and redefine the lexicographic order on the fly to mimic a sound branching strategy. Preliminary computational results are reported. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new methodology for handling optimization problems with uncertain data. With the usual Robust Optimization paradigm, one looks for the decisions ensuring a required performance for all realizations of the data from a given bounded uncertainty set, whereas with the proposed approach, we require also a controlled deterioration in performance when the data is outside the uncertainty set. The extension of Robust Optimization methodology developed in this paper opens up new possibilities to solve efficiently multi-stage finite-horizon uncertain optimization problems, in particular, to analyze and to synthesize linear controllers for discrete time dynamical systems. [ABSTRACT FROM AUTHOR]

This article presents a family of semidefinite programming bounds, obtained by Lagrangian duality, for 0-1 quadratic optimization problems with linear or quadratic constraints. These bounds have useful computational properties: they have a good ratio of tightness to computing time, they can be optimized by a quasi-Newton method, and their final tightness level is controlled by a real parameter. These properties are illustrated on three standard combinatorial optimization problems: unconstrained 0-1 quadratic optimization, heaviest $$k$$-subgraph, and graph bisection. [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]

This paper provides a detailed analysis of a primal-dual interior-point method for PDE-constrained optimization. Considered are optimal control problems with control constraints in L p . It is shown that the developed primal-dual interior-point method converges globally and locally superlinearly. Not only the easier L ∞-setting is analyzed, but also a more involved L q -analysis, q < ∞, is presented. In L ∞, the set of feasible controls contains interior points and the Fréchet differentiability of the perturbed optimality system can be shown. In the L q -setting, which is highly relevant for PDE-constrained optimization, these nice properties are no longer available. Nevertheless, a convergence analysis is developed using refined techniques. In parti- cular, two-norm techniques and a smoothing step are required. The L q -analysis with smoothing step yields global linear and local superlinear convergence, whereas the L ∞-analysis without smoothing step yields only global linear convergence. [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]

This work presents an algorithm for multiobjective optimization that is structured as: (i) a descent direction is calculated, within the cone of descent and feasible directions, and (ii) a multiobjective line search is conducted over such direction, with a new multiobjective golden section segment partitioning scheme that directly finds line-constrained efficient points that dominate the current one. This multiobjective line search procedure exploits the structure of the line-constrained efficient set, presenting a faster compression rate of the search segment than single-objective golden section line search. The proposed multiobjective optimization algorithm converges to points that satisfy the Kuhn-Tucker first-order necessary conditions for efficiency (the Pareto-critical points). Numerical results on two antenna design problems support the conclusion that the proposed method can solve robustly difficult nonlinear multiobjective problems defined in terms of computationally expensive black-box objective functions. [ABSTRACT FROM AUTHOR]

We study classic machine sequencing problems in an online setting. Specifically, we look at deterministic and randomized algorithms for the problem of scheduling jobs with release dates on identical parallel machines, to minimize the sum of weighted completion times: Both preemptive and non-preemptive versions of the problem are analyzed. Using linear programming techniques, borrowed from the single machine case, we are able to design a 2.62-competitive deterministic algorithm for the non-preemptive version of the problem, improving upon the 3.28-competitive algorithm of Megow and Schulz. Additionally, we show how to combine randomization techniques with the linear programming approach to obtain randomized algorithms for both versions of the problem with competitive ratio strictly smaller than 2 for any number of machines (but approaching two as the number of machines grows). Our algorithms naturally extend several approaches for single and parallel machine scheduling. We also present a brief computational study, for randomly generated problem instances, which suggests that our algorithms perform very well in practice. [ABSTRACT FROM AUTHOR]

I am not a historian; these are just reminiscences of a person involved in the development of optimization theory and methods in the former USSR. I realize that my point of view may be very personal; however, I am trying to present as broad and unbiased picture as I can. [ABSTRACT FROM AUTHOR]

The recent algorithm proposed by the author for convex nonsmooth optimization is tailored to the case where the objective function is additive. We highlight aspects where this specialization differs from ignoring the structure of the objective function. To assess the approach, we consider two different nonlinear multicommodity flows applications in telecommunications including some comparison with a proximal bundle algorithm. [ABSTRACT FROM AUTHOR]

We prove that the subdifferential of any semi-algebraic extended-real-valued function on $$\mathbf{R}^n$$ has $$n$$-dimensional graph. We discuss consequences for generic semi-algebraic optimization problems. [ABSTRACT FROM AUTHOR]

Bilevel programs (BL) form a special class of optimization problems. They appear in many models in economics, game theory and mathematical physics. BL programs show a more complicated structure than standard finite problems. We study the so-called KKT-approach for solving bilevel problems, where the lower level minimality condition is replaced by the KKT- or the FJ-condition. This leads to a special structured mathematical program with complementarity constraints. We analyze the KKT-approach from a generic viewpoint and reveal the advantages and possible drawbacks of this approach for solving BL problems numerically. [ABSTRACT FROM AUTHOR]

Bilevel programming problems are often reformulated using the Karush-Kuhn-Tucker conditions for the lower level problem resulting in a mathematical program with complementarity constraints(MPCC). Clearly, both problems are closely related. But the answer to the question posed is 'No' even in the case when the lower level programming problem is a parametric convex optimization problem. This is not obvious and concerns local optimal solutions. We show that global optimal solutions of the MPCC correspond to global optimal solutions of the bilevel problem provided the lower-level problem satisfies the Slater's constraint qualification. We also show by examples that this correspondence can fail if the Slater's constraint qualification fails to hold at lower-level. When we consider the local solutions, the relationship between the bilevel problem and its corresponding MPCC is more complicated. We also demonstrate the issues relating to a local minimum through examples. [ABSTRACT FROM AUTHOR]

n Adaptive Regularisation algorithm using Cubics (ARC) is proposed for unconstrained optimization, 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 et al. (Optim Methods Softw 22(3):413-431, 2007). At each iteration of our approach, an approximate global minimizer of a local cubic regularisation of the objective function is determined, and this ensures a significant improvement in the objective so long as the Hessian of the objective is locally Lipschitz continuous. The new method uses an adaptive estimation of the local Lipschitz constant and approximations to the global model-minimizer which remain computationally-viable even for large-scale problems. We show that the excellent global and local convergence properties obtained by Nesterov and Polyak are retained, and sometimes extended to a wider class of problems, by our ARC approach. Preliminary numerical experiments with small-scale test problems from the CUTEr set show encouraging performance of the ARC algorithm when compared to a basic trust-region implementation. [ABSTRACT FROM AUTHOR]

We propose necessary and sufficient conditions for a sensing matrix to be ' s-semigood' - to allow for exact ℓ-recovery of sparse signals with at most s nonzero entries under sign restrictions on part of the entries. We express error bounds for imperfect ℓ-recovery in terms of the characteristics underlying these conditions. These characteristics, although difficult to evaluate, lead to verifiable sufficient conditions for exact sparse ℓ-recovery and thus efficiently computable upper bounds on those s for which a given sensing matrix is s-semigood. We examine the properties of proposed verifiable sufficient conditions, describe their limits of performance and provide numerical examples comparing them with other verifiable conditions from the literature. [ABSTRACT FROM AUTHOR]

We introduce the concept of a Markov risk measure and we use it to formulate risk-averse control problems for two Markov decision models: a finite horizon model and a discounted infinite horizon model. For both models we derive risk-averse dynamic programming equations and a value iteration method. For the infinite horizon problem we develop a risk-averse policy iteration method and we prove its convergence. We also propose a version of the Newton method to solve a nonsmooth equation arising in the policy iteration method and we prove its global convergence. Finally, we discuss relations to min-max Markov decision models. [ABSTRACT FROM AUTHOR]

We introduce a new trust-region method for unconstrained optimization where the radius update is computed using the model information at the current iterate rather than at the preceding one. The update is then performed according to how well the current model retrospectively predicts the value of the objective function at last iterate. Global convergence to first- and second-order critical points is proved under classical assumptions and preliminary numerical experiments on CUTEr problems indicate that the new method is very competitive. [ABSTRACT FROM AUTHOR]

Given an n × n symmetric possibly indefinite matrix A, a modified Cholesky algorithm computes a factorization of the positive definite matrix A + E, where E is a correction matrix. Since the factorization is often used to compute a Newton-like downhill search direction for an optimization problem, the goals are to compute the modification without much additional cost and to keep A + E well-conditioned and close to A. Gill, Murray and Wright introduced a stable algorithm, with a bound of || E||2 = O( n 2). An algorithm of Schnabel and Eskow further guarantees || E||2 = O( n). We present variants that also ensure || E||2 = O( n). Moré and Sorensen and Cheng and Higham used the block LBL T factorization with blocks of order 1 or 2. Algorithms in this class have a worst-case cost O( n 3) higher than the standard Cholesky factorization. We present a new approach using a sandwiched LTL T - LBL T factorization, with T tridiagonal, that guarantees a modification cost of at most O( n 2). [ABSTRACT FROM AUTHOR]

By refining a variant of the Klee–Minty example that forces the central path to visit all the vertices of the Klee–Minty n-cube, we exhibit a nearly worst-case example for path-following interior point methods. Namely, while the theoretical iteration-complexity upper bound is $$O(2^{n}n^{\frac{5}{2}})$$ , we prove that solving this n-dimensional linear optimization problem requires at least 2 n −1 iterations. [ABSTRACT FROM AUTHOR]

An important question in discrete optimization under uncertainty is to understand the persistency of a decision variable, i.e., the probability that it is part of an optimal solution. For instance, in project management, when the task activity times are random, the challenge is to determine a set of critical activities that will potentially lie on the longest path. In the spanning tree and shortest path network problems, when the arc lengths are random, the challenge is to pre-process the network and determine a smaller set of arcs that will most probably be a part of the optimal solution under different realizations of the arc lengths. Building on a characterization of moment cones for single variate problems, and its associated semidefinite constraint representation, we develop a limited marginal moment model to compute the persistency of a decision variable. Under this model, we show that finding the persistency is tractable for zero-one optimization problems with a polynomial sized representation of the convex hull of the feasible region. Through extensive experiments, we show that the persistency computed under the limited marginal moment model is often close to the simulated persistency value under various distributions that satisfy the prescribed marginal moments and are generated independently. [ABSTRACT FROM AUTHOR]

We present a survey on the results related to the theory of lexicographic differentiation. This theory ensures an efficient computation of generalized ( lexicographic) derivative of a nonsmooth function belonging to a special class of lexicographically smooth functions. This class is a linear space which contains all differentiable functions, all convex functions, and which is closed with respect to component-wise composition of the members. In order to define lexicographic derivative in a unique way, it is enough to fix a basis in the space of variables. Lexicographic derivatives can be used in black-box optimization methods. We give some examples of applications of these derivatives in analysis of nonsmooth functions. It is shown that the system of lexicographic derivatives along a fixed basis correctly represents corresponding nonsmooth function (Newton-Leibnitz formula). We present nonsmooth versions of standard theorems on potentiality of nonlinear operators, on differentiation of parametric integrals and on differentiation of functional sequences. Finally, we show that an appropriately defined lexicographic subdifferential ensures a more rigorous selection of a candidate optimal solution than the subdifferential of Clarke. [ABSTRACT FROM AUTHOR]

Abstract. We consider primal-dual algorithms for certain types of infinite-dimensional optimization problems. Our approach is based on the generalization of the technique of finite dimensional Euclidean Jordan algebras to the case of infinite-dimensional JB-algebras of finite rank. This generalization enables us to develop polynomial-time primal-dual algorithms for "infinite-dimensional second-order cone programs." We consider as an example a long step primal dual algorithm based on the Nesterov-Todd direction. It is shown that this algorithm can be generalized along with complexity estimates to the infinite-dimensional situation under consideration. An application is given to an important problem of control theory: multi-criteria analytic design of the linear regulator. The calculation of the Nesterov Todd direction requires in this case solving one matrix differential Riccati equation plus solving a finite-dimensional system of linear algebraic equations on each iteration. The number of equations and unknown variables of this algebraic system is m + 1, where m is a number of quadratic performance criteria. [ABSTRACT FROM AUTHOR]

Traveling Salesman, Steiner Tree, and many other famous geometric optimization problems are NP-hard. Since we do not expect to design efficient algorithms that solve these problems optimally, researchers have tried to design approximation algorithms, which can compute a provably near-optimal solution in polynomial time.We survey such algorithms, in particular a new technique developed over the past few years that allows us to design approximation schemes for many of these problems. For any fixed constant c> 0, the algorithm can compute a solution whose cost is at most (1 + c) times the optimum. (The running time is polynomial for every fixed c> 0, and in many cases is even nearly linear.) We describe how these schemes are designed, and survey the status of a large number of problems. [ABSTRACT FROM AUTHOR]