The circumpolygon with maximal area

The convex circumpolygon with maximal area to a given convex polygon can be determined by means of dynamic programming. The effort of this method increases cubically with respect to the number of sides. It is further shown that the optimal circum-polygon can be constructed with ruler and circle. The applied version of dynamic programming can be also used for solving Steiner's problem of the inpoiygon with minimal circumference but it demands a higher effort than Phú's method.     

Branch-and-Bound Outer Approximation Algorithm for Sum-of-Ratios Fractional Programs

In this article, a branch and-bound outer approximation algorithm is presented for globally solving a sum-of-ratios fractional programming problem. To solve this problem, the algorithm instead solves an equivalent problem that involves minimizing an indefinite quadratic function over a nonempty, compact convex set. This problem is globally solved by a branch-and-bound outer approximation approach that can create several closed-form linear inequality cuts per iteration. In contrast to pure outer approximation techniques, the algorithm does not require computing the new vertices that are created as these cuts are added. Computationally, the main work of the algorithm involves solving a sequence of convex programming problems whose feasible regions are identical to one another except for certain linear constraints. As a result, to solve these problems, an optimal solution to one problem can potentially be used to good effect as a starting solution for the next problem.     

An efficient algorithm for facility location in the presence of forbidden regions

This paper investigates a constrained form of the classical Weber problem. Specifically, we consider the problem of locating a new facility in the presence of convex polygonal forbidden regions such that the sum of the weighted distances from the new facility to n existing facilities is minimized. It is assumed that a forbidden region is an area in the plane where travel and facility location are not permitted and that distance is measured using the Euclidean-distance metric. A solution procedure for this nonconvex programming problem is presented. It is shown that by iteratively solving a series of unconstrained problems, this procedure terminates at a local optimum to the original constrained problem. Numerical examples are presented.     

Generalized linear multiplicative and fractional programming

This paper addresses itself to the algorithm for minimizing the sum of a convex function and a product of two linear functions over a polytope. It is shown that this nonconvex minimization problem can be solved by solving a sequence of convex programming problems. The basic idea of this algorithm is to embed the original problem into a problem in higher dimension and apply a parametric programming (path following) approach. Also it is shown that the same idea can be applied to a generalized linear fractional programming problem whose objective function is the sum of a convex function and a linear fractional function.     

Affine minorants minimizing the sum of convex functions

It is often possible to replace a convex minimization problem by an equivalent one, in which each of the original convex functions is replaced by a suitably chosen affine minorant. In this paper we identify essentially the minimal conditions permitting this replacement, and also shed light on the close and complete link between such optimal affine minorants and certain optimal dual vectors. An application to the ordinary convex programming problem is included.This research was supported in part by the National Science Foundation, Grant No. MPS75-08025, at the University of Illinois at Urbana-Champaign.     

A finite steps algorithm for solving convex feasibility problems

This paper develops a new variant of the classical alternating projection method for solving convex feasibility problems where the constraints are given by the intersection of two convex cones in a Hilbert space. An extension to the feasibility problem for the intersection of two convex sets is presented as well. It is shown that one can solve such problems in a finite number of steps and an explicit upper bound for the required number of steps is obtained. As an application, we propose a new finite steps algorithm for linear programming with linear matrix inequality constraints. This solution is computed by solving a sequence of a matrix eigenvalue decompositions. Moreover, the proposed procedure takes advantage of the structure of the problem. In particular, it is well adapted for problems with several small size constraints.     

On the approximate solution of nonlinear variational inequalities

Summary Nonlinear locally coercive variational inequalities are considered and especially the minimal surface over an obstacle. Optimal or nearly optimal error estimates are proved for a direct discretization of the problem with linear finite elements on a regular triangulation of the not necessarily convex domain. It is shown that the solution may be computed by a globally convergent relaxation method. Some numerical results are presented.     

An algorithm for solving the problem of convex programming with several objective functions

This work aims to establish an algorithm for solving the problem of convex programming with several objective-functions, with linear constraints. Starting from the idea of Rosen’s algorithm for solving the problem of convex programming with linear constraints, and taking into account the solution concept from multidimensional programming, represented by a program which reaches ”the best compromise”, we are extending this method in the case of multidimensional programming. The concept of direction of minimization is introduced, and a necessary and sufficient condition is given for as ∈R ⁿ direction to be a direction of minimization, according to the values of a criteria ensemble in a given point. The algorithm is interactive, and the intervention of the decident is minimal. The two numerical examples presented at the end validate the algorithm.     

An exact algorithm for graph partitioning

An exact algorithm is presented for solving edge weighted graph partitioning problems. The algorithm is based on a branch and bound method applied to a continuous quadratic programming formulation of the problem. Lower bounds are obtained by decomposing the objective function into convex and concave parts and replacing the concave part by an affine underestimate. It is shown that the best affine underestimate can be expressed in terms of the center and the radius of the smallest sphere containing the feasible set. The concave term is obtained either by a constant diagonal shift associated with the smallest eigenvalue of the objective function Hessian, or by a diagonal shift obtained by solving a semidefinite programming problem. Numerical results show that the proposed algorithm is competitive with state-of-the-art graph partitioning codes.     

Convexification and Concavification for a General Class of Global Optimization Problems

A kind of general convexification and concavification methods is proposed for solving some classes of global optimization problems with certain monotone properties. It is shown that these minimization problems can be transformed into equivalent concave minimization problem or reverse convex programming problem or canonical D.C. programming problem by using the proposed convexification and concavification schemes. The existing algorithms then can be used to find the global solutions of the transformed problems.     

Using a Conic Formulation for Finding Steiner Minimal Trees

We present a new mathematical programming formulation for the Steiner minimal tree problem. We relax the integrality constraints on this formulation and transform the resulting problem (which is convex, but not everywhere differentiable) into a standard convex programming problem in conic form. We consider an efficient computation of an ε-optimal solution for this latter problem using an interior-point algorithm.     

Deriving the Properties of Linear Bilevel Programming via a Penalty Function Approach

For the linear bilevel programming problem, we propose an assumption weaker than existing assumptions, while achieving similar results via a penalty function approach. The results include: equivalence between (i) existence of a solution to the problem, (ii) existence of an exact penalty function approach for solving the problem, and (iii) achievement of the optimal value of the equivalent form of the problem at some vertex of a certain polyhedral convex set. We prove that the assumption is both necessary and sufficient for the linear bilevel programming problem to admit an exact penalty function formulation, provided that the equivalent form of the problem has a feasible solution. A method is given for computing the minimal penalty function parameter value. This method can be executed by solving a set of linear programming problems. Lagrangian duality is also presented.     

Global minimization of a generalized convex multiplicative function

This paper discusses an algorithm for generalized convex multiplicative programming problems, a special class of nonconvex minimization problems in which the objective function is expressed as a sum ofp products of two convex functions. It is shown that this problem can be reduced to a concave minimization problem with only 2p variables. An outer approximation algorithm is proposed for solving the resulting problem.     

A cutting plane method for solving minimax problems in the complex plane

It is shown how the combined discretization and cutting plane method for general convex semi-infinite programming problems recently presented in [40] can be effectively implemented for the solution of minimax problems in the complex plane. In contrast to other approaches, the minimax problem does not have to be linearized. The performance of the algorithm is demonstrated by a number of highly accurate numerical examples.     

An entropy based central cutting plane algorithm for convex min-max semi-infinite programming problems

In this paper,we present a central cutting plane algorithm for solving convex min-max semi-infinite programming problems.Because the objective function here is non-differentiable,we apply a smoothing technique to the considered problem and develop an algorithm based on the entropy function.It is shown that the global convergence of the proposed algorithm can be obtained under weaker conditions.Some numerical results are presented to show the potential of the proposed algorithm.     

Inscribed ball and enclosing box methods for the convex maximization problem

Many important classes of decision models give rise to the problem of finding a global maximum of a convex function over a convex set. This problem is known also as concave minimization, concave programming or convex maximization. Such problems can have many local maxima, therefore finding the global maximum is a computationally difficult problem, since standard nonlinear programming procedures fail. In this article, we provide a very simple and practical approach to find the global solution of quadratic convex maximization problems over a polytope. A convex function achieves its global maximum at extreme points of the feasible domain. Since an inscribed ball does not contain any extreme points of the domain, we use the largest inscribed ball for an inner approximation while a minimal enclosing box is exploited for an outer approximation of the domain. The approach is based on the use of these approximations along with the standard local search algorithm and cutting plane techniques.     

A model of distributionally robust two-stage stochastic convex programming with linear recourse

We consider distributionally robust two-stage stochastic convex programming problems, in which the recourse problem is linear. Other than analyzing these new models case by case for different ambiguity sets, we adopt a unified form of ambiguity sets proposed by Wiesemann, Kuhn and Sim, and extend their analysis from a single stochastic constraint to the two-stage stochastic programming setting. It is shown that under a standard set of regularity conditions, this class of problems can be converted to a conic optimization problem. Numerical results are presented to show the efficiency of the distributionally robust approach.     

广义多品种最小费用流问题的对偶理论

基于广义多品种最小费用流问题的性质，将问题转化成一对含有内、外层问题的双水平规划，内层规划实际是单品种费用流问题，而外层问题是分离的凸规划，使用相关的凸分析理论，导出了广义多品种最小费用流问题的对偶规划，对偶定理和Kuhn-Tucker条件。     

Unboundedness in reverse convex and concave integer programming

In this paper we are concerned with the problem of unboundedness and existence of an optimal solution in reverse convex and concave integer optimization problems. In particular, we present necessary and sufficient conditions for existence of an upper bound for a convex objective function defined over the feasible region contained in ${\mathbb{Z}^n}$ . The conditions for boundedness are provided in a form of an implementable algorithm, showing that for the considered class of functions, the integer programming problem is unbounded if and only if the associated continuous problem is unbounded. We also address the problem of boundedness in the global optimization problem of maximizing a convex function over a set of integers contained in a convex and unbounded region. It is shown in the paper that in both types of integer programming problems, the objective function is either unbounded from above, or it attains its maximum at a feasible integer point.     

Constructive Solution of Inverse Parametric Linear/Quadratic Programming Problems

Parametric convex programming has received a lot of attention, since it has many applications in chemical engineering, control engineering, signal processing, etc. Further, inverse optimality plays an important role in many contexts, e.g., image processing, motion planning. This paper introduces a constructive solution of the inverse optimality problem for the class of continuous piecewise affine functions. The main idea is based on the convex lifting concept. Accordingly, an algorithm to construct convex liftings of a given convexly liftable partition will be put forward. Following this idea, an important result will be presented in this article: Any continuous piecewise affine function defined over a polytopic partition is the solution of a parametric linear/quadratic programming problem. Regarding linear optimal control, it will be shown that any continuous piecewise affine control law can be obtained via a linear optimal control problem with the control horizon at most equal to 2 prediction steps.