Robust duality for generalized convex programming problems under data uncertainty

In this paper we present a robust duality theory for generalized convex programming problems in the face of data uncertainty within the framework of robust optimization. We establish robust strong duality for an uncertain nonlinear programming primal problem and its uncertain Lagrangian dual by showing strong duality between the deterministic counterparts: robust counterpart of the primal model and the optimistic counterpart of its dual problem. A robust strong duality theorem is given whenever the Lagrangian function is convex. We provide classes of uncertain non-convex programming problems for which robust strong duality holds under a constraint qualification. In particular, we show that robust strong duality is guaranteed for non-convex quadratic programming problems with a single quadratic constraint with the spectral norm uncertainty under a generalized Slater condition. Numerical examples are given to illustrate the nature of robust duality for uncertain nonlinear programming problems. We further show that robust duality continues to hold under a weakened convexity condition.     

Global optimization of a class of nonconvex quadratically constrained quadratic programming problems

In this paper we study a class of nonconvex quadratically constrained quadratic programming problems generalized from relaxations of quadratic assignment problems. We show that each problem is polynomially solved. Strong duality holds if a redundant constraint is introduced. As an application, a new lower bound is proposed for the quadratic assignment problem.     

A Branch-and-Bound Approach for Solving a Class of Generalized Semi-infinite Programming Problems

A nonconvex generalized semi-infinite programming problem is considered, involving parametric max-functions in both the objective and the constraints. For a fixed vector of parameters, the values of these parametric max-functions are given as optimal values of convex quadratic programming problems. Assuming that for each parameter the parametric quadratic problems satisfy the strong duality relation, conditions are described ensuring the uniform boundedness of the optimal sets of the dual problems w.r.t. the parameter. Finally a branch-and-bound approach is suggested transforming the problem of finding an approximate global minimum of the original nonconvex optimization problem into the solution of a finite number of convex problems.     

A derivative algorithm for inexact quadratic program – application to environmental decision-making under uncertainty

Inexact quadratic programming (IQP) is an extension of conventional quadratic programming for handling both nonlinearities in cost objectives and uncertainties with modeling parameters. It has been a useful tool for environmental systems analysis. However, inefficiency in its solution method has existed, leading to difficulties in its practical application. In this study, a derivative algorithm (DAM) is proposed for solving the IQP. It improves upon the existing method through provision of a quantitative expression for uncertain relationships between the quadratic objective function and the decision variables. The DAM requires much lower computational efforts than the existing algorithm, which is especially meaningful for the IQP's application to large-scale problems. The developed DAM is applied to a hypothetical problem of municipal solid waste management and planning. Detailed solution steps are provided to clearly demonstrate the method's advantages.     

Coradiant sets and $$\varepsilon $$-efficiency in multiobjective optimization

In multi-parametric programming an optimization problem is solved as a function of certain parameters, where the parameters are commonly considered to be bounded and continuous. In this paper, we use the case of strictly convex multi-parametric quadratic programming (mp-QP) problems with affine constraints to investigate problems where these conditions are not met. Based on the combinatorial solution approach for mp-QP problems featuring bounded and continuous parameters, we show that (i) for unbounded parameters, it is possible to obtain the multi-parametric solution if there exists one realization of the parameters for which the optimization problem can be solved and (ii) for binary parameters, we present the equivalent mixed-integer formulations for the application of the combinatorial algorithm. These advances are combined into a new, generalized version of the combinatorial algorithm for mp-QP problems, which enables the solution of problems featuring both unbounded and binary parameters. This novel approach is applied to mixed-integer bilevel optimization problems and the parametric solution of the dual of a convex problem.     

Parallel interior-point solver for structured quadratic programs: Application to financial planning problems

Many practical large-scale optimization problems are not only sparse, but also display some form of block-structure such as primal or dual block angular structure. Often these structures are nested: each block of the coarse top level structure is block-structured itself. Problems with these characteristics appear frequently in stochastic programming but also in other areas such as telecommunication network modelling. We present a linear algebra library tailored for problems with such structure that is used inside an interior point solver for convex quadratic programming problems. Due to its object-oriented design it can be used to exploit virtually any nested block structure arising in practical problems, eliminating the need for highly specialised linear algebra modules needing to be written for every type of problem separately. Through a careful implementation we achieve almost automatic parallelisation of the linear algebra. The efficiency of the approach is illustrated on several problems arising in the financial planning, namely in the asset and liability management. The problems are modelled as multistage decision processes and by nature lead to nested block-structured problems. By taking the variance of the random variables into account the problems become non-separable quadratic programs. A reformulation of the problem is proposed which reduces density of matrices involved and by these means significantly simplifies its solution by an interior point method. The object-oriented parallel solver achieves high efficiency by careful exploitation of the block sparsity of these problems. As a result a problem with over 50 million decision variables is solved in just over 2 hours on a parallel computer with 16 processors. The approach is by nature scalable and the parallel implementation achieves nearly perfect speed-ups on a range of problems. Supported by the Engineering and Physical Sciences Research Council of UK, EPSRC grant GR/R99683/01     

一种非增值型凸二次双层规划的有效算法

主要研究了非增值型凸二次双层规划的一种有效求解算法。首先利用数学规划的对偶理论,将所求双层规划转化为一个下层只有一个无约束凸二次子规划的双层规划问题.然后根据两个双层规划的最优解和最优目标值之间的关系,提出一种简单有效的算法来解决非增值型凸二次双层规划问题.并通过数值算例的计算结果说明了该算法的可行性和有效性。     

Parametric global optimisation for bilevel programming

We propose a global optimisation approach for the solution of various classes of bilevel programming problems (BLPP) based on recently developed parametric programming algorithms. We first describe how we can recast and solve the inner (follower’s) problem of the bilevel formulation as a multi-parametric programming problem, with parameters being the (unknown) variables of the outer (leader’s) problem. By inserting the obtained rational reaction sets in the upper level problem the overall problem is transformed into a set of independent quadratic, linear or mixed integer linear programming problems, which can be solved to global optimality. In particular, we solve bilevel quadratic and bilevel mixed integer linear problems, with or without right-hand-side uncertainty. A number of examples are presented to illustrate the steps and details of the proposed global optimisation strategy.     

Computational strategies for non-convex multistage MINLP models with decision-dependent uncertainty and gradual uncertainty resolution

In many planning problems under uncertainty the uncertainties are decision-dependent and resolve gradually depending on the decisions made. In this paper, we address a generic non-convex MINLP model for such planning problems where the uncertain parameters are assumed to follow discrete distributions and the decisions are made on a discrete time horizon. In order to account for the decision-dependent uncertainties and gradual uncertainty resolution, we propose a multistage stochastic programming model in which the non-anticipativity constraints in the model are not prespecified but change as a function of the decisions made. Furthermore, planning problems consist of several scenario subproblems where each subproblem is modeled as a nonconvex mixed-integer nonlinear program. We propose a solution strategy that combines global optimization and outer-approximation in order to optimize the planning decisions. We apply this generic problem structure and the proposed solution algorithm to several planning problems to illustrate the efficiency of the proposed method with respect to the method that uses only global optimization.     

On the global optimality of generalized trust region subproblems

Quadratically constrained quadratic programming is an important class of optimization problems. We consider the case with one quadratic constraint. Since both the objective function and its constraint can be neither convex nor concave, it is also known as the ‘generalized trust region subproblem.’ The theory and algorithms for this problem have been well studied under the Slater condition. In this article, we analyse the duality property between the primal problem and its Lagrangian dual problem, and discuss the attainability of the optimal primal solution without the Slater condition. The relations between the Lagrangian dual and semidefinite programming dual is also given.     

对带有盒约束的二次整数规划的一种线性化方法

本文主要讨论了二次整数规划问题的线性化方法.在目标函数为二次函数的情况下,我们讨论了带有二次约束的整数规划问题的线性化方法,并将文献中对二次0-1问题的研究拓展为对带有盒约束的二次整数规划问题的研究.最终将带有盒约束的二次整数规划问题转化为线性混合本文主要讨论了二次整数规划问题的线性化方法.在目标函数为二次函数的情况下,我们讨论了带有二次约束的整数规划问题的线性化方法,并将文献中对二次0-1问题的研究拓展为对带有盒约束的二次整数规划问题的研究.最终将带有盒约束的二次整数规划问题转化为线性混合0-1整数规划问题,然后利用Ilog-cplex或Excel软件中的规划求解工具进行求解,从而解决原二次整数规划.     

Parallel distributed block coordinate descent methods based on pairwise comparison oracle

The aim of this paper is to find the global solutions of uncertain optimization problems having a quadratic objective function and quadratic inequality constraints. The bounded epistemic uncertainties in the constraint coefficients are represented using either universal or existential quantified parameters and interval parameter domains. This approach allows to model non-controlled uncertainties by using universally quantified parameters and controlled uncertainties by using existentially quantified ones. While existentially quantified parameters could be equivalently considered as additional variables, keeping them as parameters allows maintaining the quadratic problem structure, which is essential for the proposed algorithm. The branch and bound algorithm presented in the paper handles both universally and existentially quantified parameters in a homogeneous way, without branching on their domains, and uses some dedicated numerical constraint programming techniques for finding a robust, global solution. Several examples clarify the theoretical parts and the tests demonstrate the usefulness of the proposed method.     

A two-phase algorithm for the partial accessibility constrained vehicle routing problem

In the partial accessibility constrained vehicle routing problem, a route can be covered by two types of vehicles, i.e. truck or truck + trailer. Some customers are accessible by both vehicle types, whereas others solely by trucks. After introducing an integer programming formulation for the problem, we describe a two-phase heuristic method which extends a classical vehicle routing algorithm. Since it is necessary to solve a combinatorial problem that has some similarities with the generalized assignment problem, we propose an enumerative procedure in which bounds are obtained from a Lagrangian relaxation. The routine provides very encouraging results on a set of test problems.     

Parametric simplex algorithms for solving a special class of nonconvex minimization problems

It is shown that parametric linear programming algorithms work efficiently for a class of nonconvex quadratic programming problems called generalized linear multiplicative programming problems, whose objective function is the sum of a linear function and a product of two linear functions. Also, it is shown that the global minimum of the sum of the two linear fractional functions over a polytope can be obtained by a similar algorithm. Our numerical experiments reveal that these problems can be solved in much the same computational time as that of solving associated linear programs. Furthermore, we will show that the same approach can be extended to a more general class of nonconvex quadratic programming problems.     

Generalized proximal point algorithm for convex optimization

Ha (Ref. 1) recently introduced a generalized proximal point algorithm for solving a generalized equation. In this note, we present a generalized proximal point algorithm for convex optimization problems based on Ha's work. The idea behind this algorithm is that, instead of adding a quadratic term to all the variables, we add a quadratic term to a subset of the variables. We extend the criteria for approximate solutions given by Rockafellar (Ref. 2) and Auslender (Ref. 3) and present convergence results. Finally, we show how this algorithm can be applied to solve block-angular linear and quadratic programming problems.     

On fuzzy random multiobjective quadratic programming

In this paper, a multiobjective quadratic programming problem having fuzzy random coefficients matrix in the objective and constraints and the decision vector are fuzzy pseudorandom variables is considered. First, we show that the efficient solutions of fuzzy quadratic multiobjective programming problems are resolved into series-optimal-solutions of relative scalar fuzzy quadratic programming. Some theorems are proved to find an optimal solution of the relative scalar quadratic multiobjective programming with fuzzy coefficients, having decision vectors as fuzzy variables. At the end, numerical examples are illustrated in the support of the obtained results.     

Optimality conditions and duality models for generalized fractional programming problems containing locally subdifferentiable and ρ:-convex functions

Both parametric and nonparametric necessary and sufficient optimality conditions are established for a class of nonsmooth generalized fractional programming problems containing ρ-convex functions. Subsequently, these optimality criteria are utilized as a basis for constructing two parametric and four parameter-free duality models and proving appropriate duality theorems. Several classes of generalized fractional programming problems, including those with arbitrary norms, square roots of positive semidefinite quadratic forms, support functions, continuous max functions, and discrete max functions, which can be viewed as special cases of the main problem are briefly discussed. The optimality and duality results developed here also contain, as special cases, similar results for nonsmooth problems with fractional, discrete max, and conventional objective functions which are particular cases of the main problem considered in this paper     

Some properties of the lower bound of optimal values in interval convex quadratic programming

One of the fundamental problems in interval quadratic programming is to compute the range of optimal values. In this paper, we derive some results on the lower bound of interval convex quadratic programming. We first develop complementary slackness conditions of a quadratic program and its Dorn dual. Then, some interesting and useful characteristics of the lower bound of interval quadratic programming are established based on these conditions. Finally, illustrative examples and remarks are given to get an insight into the problem discussed.     

Detecting copositivity of a symmetric matrix by an adaptive ellipsoid-based approximation scheme

It is co-NP-complete to decide whether a given matrix is copositive or not. In this paper, this decision problem is transformed into a quadratic programming problem, which can be approximated by solving a sequence of linear conic programming problems defined on the dual cone of the cone of nonnegative quadratic functions over the union of a collection of ellipsoids. Using linear matrix inequalities (LMI) representations, each corresponding problem in the sequence can be solved via semidefinite programming. In order to speed up the convergence of the approximation sequence and to relieve the computational effort of solving linear conic programming problems, an adaptive approximation scheme is adopted to refine the union of ellipsoids. The lower and upper bounds of the transformed quadratic programming problem are used to determine the copositivity of the given matrix.     

线性等式约束多目标规划的一个降维算法

本文提出具有线性等式约束多目标规划问题的一个降维算法．当目标函数全是二次或线性但至少有一个二次型时，用线性加权法转化原问题为单目标二次规划，再用降维方法转化为求解一个线性方程组．若目标函数非上述情形，首先用线性加权法将原问题转化为具有线性等式约束的非线性规划，然后，对这一非线性规划的目标函数二次逼近，构成线性等式约束二次规划序列，用降维法求解，直到满足精度要求为止．