Index branch-and-bound algorithm for Lipschitz univariate global optimization with multiextremal constraints

In this paper, Lipschitz univariate constrained global optimization problems where both the objective function and constraints can be multiextremal are considered. The constrained problem is reduced to a discontinuous unconstrained problem by the index scheme without introducing additional parameters or variables. A Branch-and-Bound method that does not use derivatives for solving the reduced problem is proposed. The method either determines the infeasibility of the original problem or finds lower and upper bounds for the global solution. Not all the constraints are evaluated during every iteration of the algorithm, providing a significant acceleration of the search. Convergence conditions of the new method are established. Extensive numerical experiments are presented.     

Conditions for boundedness in concave programming under reverse convex and convex constraints

In this paper, we are concerned with the problem of boundedness in the constrained global maximization of a convex function. In particular, we present necessary and sufficient conditions for boundedness of a feasible region defined by reverse convex constraints and we establish sufficient and necessary conditions for existence of an upper bound for a convex objective function defined over the system of concave inequality constraints. We also address the problem of boundedness in the global maximization problem when a feasible region is convex and unbounded.     

An exact penalty approach for solving a class of minimization problems with boolean variables

An exact penalty approach for solving minimization problems with a concave objective function, linear constraints and Boolean variables is proposed. The penalty problems have continuous variables. An estimation of the penalty parameter which guarantees the exactness can be calculated on the base of an auxiliary problem. The results are applied to problems with an arbitrary quadratic objective function, linear constraints and Boolean variables. This leads to a modified Lagrangean approach for the latter problems. In the general case, the penalty approach is compared with a direct application of results of global optimization to a modification of the initial problem.     

线性比式和规划问题的输出空间分支定界算法

本文旨在针对线性比式和规划这一NP-Hard非线性规划问题提出新的全局优化算法.首先,通过引入p个辅助变量把原问题等价的转化为一个非线性规划问题,这个非线性规划问题的目标函数是乘积和的形式并给原问题增加了p个新的非线性约束,再通过构造凸凹包络的技巧对等价问题的目标函数和约束条件进行相应的线性放缩,构成等价问题的一个下界线性松弛规划问题,从而提出了一个求解原问题的分支定界算法,并证明了算法的收敛性.最后,通过数值结果比较表明所提出的算法是可行有效的.     

A one-dimensional local tuning algorithm for solving GO problems with partially defined constraints

Lipschitz one-dimensional constrained global optimization (GO) problems where both the objective function and constraints can be multiextremal and non-differentiable are considered in this paper. Problems, where the constraints are verified in a priori given order fixed by the nature of the problem are studied. Moreover, if a constraint is not satisfied at a point, then the remaining constraints and the objective function can be undefined at this point. The constrained problem is reduced to a discontinuous unconstrained problem by the index scheme without introducing additional parameters or variables. A new geometric method using adaptive estimates of local Lipschitz constants is introduced. The estimates are calculated by using the local tuning technique proposed recently. Numerical experiments show quite a satisfactory performance of the new method in comparison with the penalty approach and a method using a priori given Lipschitz constants.This research was supported by the following grants: FIRB RBNE01WBBB, FIRB RBAU01JYPN, PRIN 2005017083-002, and RFBR 04-01-00455-a. The authors would like to thank anonymous referees for their subtle suggestions.     

Value-Estimation Function Method for Constrained Global Optimization

A novel value-estimation function method for global optimization problems with inequality constraints is proposed in this paper. The value-estimation function formulation is an auxiliary unconstrained optimization problem with a univariate parameter that represents an estimated optimal value of the objective function of the original optimization problem. A solution is optimal to the original problem if and only if it is also optimal to the auxiliary unconstrained optimization with the parameter set at the optimal objective value of the original problem, which turns out to be the unique root of a basic value-estimation function. A logarithmic-exponential value-estimation function formulation is further developed to acquire computational tractability and efficiency. The optimal objective value of the original problem as well as the optimal solution are sought iteratively by applying either a generalized Newton method or a bisection method to the logarithmic-exponential value-estimation function formulation. The convergence properties of the solution algorithms guarantee the identification of an approximate optimal solution of the original problem, up to any predetermined degree of accuracy, within a finite number of iterations.     

Index information algorithm with local tuning for solving multidimensional global optimization problems with multiextremal constraints

 Multidimensional optimization problems where the objective function and the constraints are multiextremal non-differentiable Lipschitz functions (with unknown Lipschitz constants) and the feasible region is a finite collection of robust nonconvex subregions are considered. Both the objective function and the constraints may be partially defined. To solve such problems an algorithm is proposed, that uses Peano space-filling curves and the index scheme to reduce the original problem to a H?lder one-dimensional one. Local tuning on the behaviour of the objective function and constraints is used during the work of the global optimization procedure in order to accelerate the search. The method neither uses penalty coefficients nor additional variables. Convergence conditions are established. Numerical experiments confirm the good performance of the technique. Received: April 2002 / Accepted: December 2002 Published online: March 21, 2003 RID="⋆" ID="⋆" This research was supported by the following grants: FIRB RBNE01WBBB, FIRB RBAU01JYPN, and RFBR 01–01–00587. Key Words. global optimization – multiextremal constraints – local tuning – index approach     

A global optimization algorithm for polynomial programming problems using a Reformulation-Linearization Technique

This paper is concerned with the development of an algorithm to solve continuous polynomial programming problems for which the objective function and the constraints are specified polynomials. A linear programming relaxation is derived for the problem based on a Reformulation Linearization Technique (RLT), which generates nonlinear (polynomial) implied constraints to be included in the original problem, and subsequently linearizes the resulting problem by defining new variables, one for each distinct polynomial term. This construct is then used to obtain lower bounds in the context of a proposed branch and bound scheme, which is proven to converge to a global optimal solution. A numerical example is presented to illustrate the proposed algorithm.     

Global Solution Approach for a Nonconvex MINLP Problem in Product Portfolio Optimization

The rigorous and efficient determination of the global solution of a nonconvex MINLP problem arising from product portfolio optimization introduced by Kallrath (2003) is addressed. The objective of the optimization problem is to determine the optimal number and capacity of reactors satisfying the demand and leading to a minimal total cost. Based on the model developed by Kallrath (2003), an improved formulation is proposed, which consists of a concave objective function and linear constraints with binary and continuous variables. A variety of techniques are developed to tighten the model and accelerate the convergence to the optimal solution. A customized branch and bound approach that exploits the special mathematical structure is proposed to solve the model to global optimality. Computational results for two case studies are presented. In both case studies, the global solutions are obtained and proved optimal very efficiently in contrast to available commercial MINLP solvers.     

A new global optimization method for univariate constrained twice-differentiable NLP problems

In this paper, a new global optimization method is proposed for an optimization problem with twice-differentiable objective and constraint functions of a single variable. The method employs a difference of convex underestimator and a convex cut function, where the former is a continuous piecewise concave quadratic function, and the latter is a convex quadratic function. The main objectives of this research are to determine a quadratic concave underestimator that does not need an iterative local optimizer to determine the lower bounding value of the objective function and to determine a convex cut function that effectively detects infeasible regions for nonconvex constraints. The proposed method is proven to have a finite ε-convergence to locate the global optimum point. The numerical experiments indicate that the proposed method competes with another covering method, the index branch-and-bound algorithm, which uses the Lipschitz constant.     

Deterministic global optimization with partition sets whose feasibility is not known: Application to concave minimization,reverse convex constraints,DC-programming,and Lipschitzian optimization

A crucial problem for many global optimization methods is how to handle partition sets whose feasibility is not known. This problem is solved for broad classes of feasible sets including convex sets, sets defined by finitely many convex and reverse convex constraints, and sets defined by Lipschitzian inequalities. Moreover, a fairly general theory of bounding is presented and applied to concave objective functions, to functions representable as differences of two convex functions, and to Lipschitzian functions. The resulting algorithms allow one to solve any global optimization problem whose objective function is of one of these forms and whose feasible set belongs to one of the above classes. In this way, several new fields of optimization are opened to the application of global methods.     

A nonlinear interval number programming method for uncertain optimization problems

In this paper, a method is suggested to solve the nonlinear interval number programming problem with uncertain coefficients both in nonlinear objective function and nonlinear constraints. Based on an order relation of interval number, the uncertain objective function is transformed into two deterministic objective functions, in which the robustness of design is considered. Through a modified possibility degree, the uncertain inequality and equality constraints are changed to deterministic inequality constraints. The two objective functions are converted into a single-objective problem through the linear combination method, and the deterministic inequality constraints are treated with the penalty function method. The intergeneration projection genetic algorithm is employed to solve the finally obtained deterministic and non-constraint optimization problem. Two numerical examples are investigated to demonstrate the effectiveness of the present method.     

Univariate Global Optimization with Multiextremal Non-Differentiable Constraints Without Penalty Functions

This paper proposes a new algorithm for solving constrained global optimization problems where both the objective function and constraints are one-dimensional non-differentiable multiextremal Lipschitz functions. Multiextremal constraints can lead to complex feasible regions being collections of isolated points and intervals having positive lengths. The case is considered where the order the constraints are evaluated is fixed by the nature of the problem and a constraint i is defined only over the set where the constraint i−1 is satisfied. The objective function is defined only over the set where all the constraints are satisfied. In contrast to traditional approaches, the new algorithm does not use any additional parameter or variable. All the constraints are not evaluated during every iteration of the algorithm providing a significant acceleration of the search. The new algorithm either finds lower and upper bounds for the global optimum or establishes that the problem is infeasible. Convergence properties and numerical experiments showing a nice performance of the new method in comparison with the penalty approach are given. This research was supported by the following grants: FIRB RBNE01WBBB, FIRB RBAU01JYPN, and RFBR 04-01-00455-a. The author thanks Prof. D. Grimaldi for proposing the application discussed in the paper. An erratum to this article is available at .     

Successive Optimization Method via Parametric Monotone Composition Formulation*

In this paper a successive optimization method for solving inequality constrained optimization problems is introduced via a parametric monotone composition reformulation. The global optimal value of the original constrained optimization problem is shown to be the least root of the optimal value function of an auxiliary parametric optimization problem, thus can be found via a bisection method. The parametric optimization subproblem is formulated in such a way that it is a one-parameter problem and its value function is a monotone composition function with respect to the original objective function and the constraints. Various forms can be taken in the parametric optimization problem in accordance with a special structure of the original optimization problem, and in some cases, the parametric optimization problems are convex composite ones. Finally, the parametric monotone composite reformulation is applied to study local optimality.     

Second-Order Global Optimality Conditions for Optimization Problems

Second-order optimality conditions are studied for the constrained optimization problem where the objective function and the constraints are compositions of convex functions and twice strictly differentiable functions. A second-order sufficient condition of a global minimizer is obtained by introducing a generalized representation condition. Second-order minimizer characterizations for a convex program and a linear fractional program are derived using the generalized representation condition     

AN INTERVAL ALGORITHM FOR CONSTRAINED GLOBAL OPTIMIZATION

In order to solve the constrained global optimization problem,we use penalty functions not only on constraints but also on objective function. Then within the framework of interval analysis,an interval Branch-and-Bound algorithm is given,which does not need to solve a sequence of unconstrained problems. Global convergence is proved. Numerical examples show that this algorithm is efficient.     

Second-Order Global Optimality Conditions for Optimization Problems

Second-order optimality conditions are studied for the constrained optimization problem where the objective function and the constraints are compositions of convex functions and twice strictly differentiable functions. A second-order sufficient condition of a global minimizer is obtained by introducing a generalized representation condition. Second-order minimizer characterizations for a convex program and a linear fractional program are derived using the generalized representation condition     

A modified reduced gradient method for a class of nondifferentiable problems

The class of nondifferentiable problems treated in this paper constitutes the dual of a class of convex differentiable problems. The primal problem involves faithfully convex functions of linear mappings of the independent variables in the objective function and in the constraints. The points of the dual problem where the objective function is nondifferentiable are known: the method presented here takes advantage of this fact to propose modifications necessary in the reduced gradient method to guarantee convergence.     

Column enumeration based decomposition techniques for a class of non-convex MINLP problems

We propose a decomposition algorithm for a special class of nonconvex mixed integer nonlinear programming problems which have an assignment constraint. If the assignment decisions are decoupled from the remaining constraints of the optimization problem, we propose to use a column enumeration approach. The master problem is a partitioning problem whose objective function coefficients are computed via subproblems. These problems can be linear, mixed integer linear, (non-)convex nonlinear, or mixed integer nonlinear. However, the important property of the subproblems is that we can compute their exact global optimum quickly. The proposed technique will be illustrated solving a cutting problem with optimum nonlinear programming subproblems.     

Global optimization of concave functions subject to quadratic constraints: An application in nonlinear bilevel programming

When the follower's optimality conditions are both necessary and sufficient, the nonlinear bilevel program can be solved as a global optimization problem. The complementary slackness condition is usually the complicating constraint in such problems. We show how this constraint can be replaced by an equivalent system of convex and separable quadratic constraints. In this paper, we propose different methods for finding the global minimum of a concave function subject to quadratic separable constraints. The first method is of the branch and bound type, and is based on rectangular partitions to obtain upper and lower bounds. Convergence of the proposed algorithm is also proved. For computational purposes, different procedures that accelerate the convergence of the proposed algorithm are analysed. The second method is based on piecewise linear approximations of the constraint functions. When the constraints are convex, the problem is reduced to global concave minimization subject to linear constraints. In the case of non-convex constraints, we use zero-one integer variables to linearize the constraints. The number of integer variables depends only on the concave parts of the constraint functions.Parts of the present paper were prepared while the second author was visiting Georgia Tech and the University of Florida.