A finite cutting plane procedure is proposed for generating feasible points for the extreme point mathematical programming (EPMP) problem. It is demonstrated that the proposed procedure can be used for the solution o...
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A finite cutting plane procedure is proposed for generating feasible points for the extreme point mathematical programming (EPMP) problem. It is demonstrated that the proposed procedure can be used for the solution of nonconvex programs of other types as well. A finite method is described for the EPMP problem, and computational experience on EPMP and concave minimization problems is included. Computational experience demonstrates that, using certain stragegies, the Majthay-Whinston (1974) concave minimization cutting plane technique may be significantly more effective than that of Tuy (1964). The implication is that the use of the original vertex finding procedures at solving special nonconvex programming problems is interesting both from the point of view of finiteness and also from the possibility of speeding up existing cutting plane techniques.
A modification of Tuy's cone splitting algorithm for minimizing a concave function subject to linear inequality constraints is shown to be convergent by demonstrating that the limit of a sequence of constructed co...
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A modification of Tuy's cone splitting algorithm for minimizing a concave function subject to linear inequality constraints is shown to be convergent by demonstrating that the limit of a sequence of constructed convex polytopes contains the feasible region. No geometric tolerance parameters are required.
It has recently been established by Wang and Xia that local minimizers of perimeter within a ball subject to a volume constraint must be spherical caps or planes through the origin. This verifies a conjecture of the a...
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It has recently been established by Wang and Xia that local minimizers of perimeter within a ball subject to a volume constraint must be spherical caps or planes through the origin. This verifies a conjecture of the authors and is in contrast to the situation of area-minimizing surfaces with prescribed boundary where singularities can be present in high dimensions. This result lends support to the more general conjecture that volume-constrained minimizers in arbitrary convex sets may enjoy better regularity properties than their boundary-prescribed cousins. Here, we show the importance of the convexity condition by exhibiting a simple example, given by the Simons cone, of a singular volume-constrained locally area-minimizing surface within a nonconvex domain that is arbitrarily close to the unit ball.
In partitioning methods such as branch and bound for solving global optimization problems, the so-called bisection of simplices and hyperrectangles is used since almost 40 years. Bisections are also of interest in fin...
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In partitioning methods such as branch and bound for solving global optimization problems, the so-called bisection of simplices and hyperrectangles is used since almost 40 years. Bisections are also of interest in finite-element methods. However, as far as we know, no proof has been given of the optimality of bisections with respect to other partitioning strategies. In this paper, after generalizing the current definition of partition slightly, we show that bisection is not optimal. Hybrid approaches combining different subdivision strategies with inner and outer approximation techniques can be more efficient. Even partitioning a polytope into simplices has important applications, for example in computational convexity, when one wants to find the inequality representation of a polytope with known vertices. Furthermore, a natural approach to the computation of the volume of a polytope P is to generate a simplex partition of P, since the volume of a simplex is given by a simple formula. We propose several variants of the partitioning rules and present complexity considerations. Finally, we discuss an approach for the volume computation of so-called H-polytopes, i.e., polytopes given by a system of affine inequalities. Upper bounds for the number of iterations are presented and advantages as well as drawbacks are discussed.
In this paper, global optimization of linear programs with an additional reverse convex constraint is considered. This type of problem arises in many applications such as engineering design, communications networks, a...
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In this paper, global optimization of linear programs with an additional reverse convex constraint is considered. This type of problem arises in many applications such as engineering design, communications networks, and many management decision support systems with budget constraints and economies-of-scale. The main difficulty with this type of problem is the presence of the complicated reverse convex constraint., which destroys the convexity and possibly the connectivity of the feasible region, putting the problem in a class of difficult and mathematically intractable problems. We present a cutting plane method within the scope of a branch-and-bound scheme that efficiently partitions the polytope associated with the linear constraints and systematically fathoms these portions through the use of the bounds. An upper bound and a lower bound for the optimal value is found and improved at each iteration. The algorithm terminates when all the generated subdivisions have been fathomed.
We consider a class of problems of resource allocation under economies of scale, namely that of minimizing a lower semicontinuous, isotone, and explicitly quasiconcave cost function subject to linear constraints. An i...
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We consider a class of problems of resource allocation under economies of scale, namely that of minimizing a lower semicontinuous, isotone, and explicitly quasiconcave cost function subject to linear constraints. An important class of algorithms for the linearly constrained minimization of nonconvex cost functions utilize the branch and bound approach, using convex underestimating cost functions to compute the lower *** suggest instead the use of the surrogate dual problem to bound subproblems. We show that the success of the surrogate dual in fathoming subproblems in a branch and bound algorithm may be determined without directly solving the surrogate dual itself, but that a simple test of the feasibility of a certain linear system of inequalities will suffice. This test is interpreted geometrically and used to characterize the extreme points and extreme rays of the optimal value function's level sets.
For fractional programs involving several ratios in the objective function, a dual is introduced with the help of Farkas' lemma. Often the dual is again a generalized fractional program. Duality relations are esta...
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For fractional programs involving several ratios in the objective function, a dual is introduced with the help of Farkas' lemma. Often the dual is again a generalized fractional program. Duality relations are established under weak assumptions. This is done in both the linear case and the nonlinear case. We show that duality can be obtained for these nonconvex programs using only a basic result on linear (convex) inequalities.
As is well known, a saddle point for the Lagrangian function, if it exists, provides a solution to a convex programming problem; then, the values of the optimal primal and dual objective functions are equal. However, ...
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As is well known, a saddle point for the Lagrangian function, if it exists, provides a solution to a convex programming problem; then, the values of the optimal primal and dual objective functions are equal. However, these results are not valid for nonconvex problems. In this paper, several results are presented on the theory of the generalized Lagrangian function, extended from the classical Lagrangian and the generalized duality program. Theoretical results for convex problems also hold for nonconvex problems by extension of the Lagrangian function. The concept of supporting hypersurfaces is useful to add a geometric interpretation to computational algorithms. This provides a basis to develop a new algorithm.
Problems that can be reduced to polynomial and parametrized linear matrix inequalities are considered. Such problems arise, for example, in control theory. Well-known methods for their solution based on a search for n...
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Problems that can be reduced to polynomial and parametrized linear matrix inequalities are considered. Such problems arise, for example, in control theory. Well-known methods for their solution based on a search for nonnegative polynomials scale poorly and require significant computational resources. An approach based on systematic transformations of the problem under study to a form that can be addressed with simpler methods is presented.
This paper proposes prediction-and-sensing-based spectrum sharing, a new spectrum-sharing model for cognitive radio networks, with a time structure for each resource block divided into a spectrum prediction-and-sensin...
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This paper proposes prediction-and-sensing-based spectrum sharing, a new spectrum-sharing model for cognitive radio networks, with a time structure for each resource block divided into a spectrum prediction-and-sensing phase and a data transmission phase. Cooperative spectrum prediction is incorporated as a sub-phase of spectrum sensing in the first phase. We investigate a joint design of transmit beamforming at the secondary base station (BS) and sensing time. The primary design goal is to maximize the sum rate of all secondary users (SUs) subject to the minimum rate requirement for all SUs, the transmit power constraint at the secondary BS, and the interference power constraints at all primary users. The original problem is difficult to solve since it is highly nonconvex. We first convert the problem into a more tractable form, then arrive at a convex program based on an inner approximation framework, and finally propose a new algorithm to successively solve this convex program. We prove that the proposed algorithm iteratively improves the objective while guaranteeing convergence at least to local optima. Simulation results demonstrate that the proposed algorithm reaches a stationary point after only a few iterations with a substantial performance improvement over existing approaches.
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