formulating the mixed integer fractional posynomial programming the mixed integer fractional posynomial programming mifpp problem arises from the summation minimization of several quotient terms appearing in the objective function subject to given constraints all decision variables of the problem could be binary integer and or continuous with without absolute value functions this paper addresses the problem of providing an optimization approach which could certainly obtain a solution as close as possible to a global optimum in order to solve the complex mifpp problem several strategies are used including linear programming relaxation modified goal programming logarithmic piecewise function lpf and so on in addition the proposed model solves the undefined problem of the logarithm of zero or negative in the lpf this significantly improved the utility of lpf in real applications finally an illustrative example is included to demonstrate the solution procedure of the proposed model

cauchy like and pellet like results for polynomials abstract we obtain several cauchy like and pellet like results for the zeros of a general complex polynomial by considering similarity transformations of the squared companion matrix and by treating the zeros of a scalar polynomial as the eigenvalues of a matrix polynomial

convergence of interval type algorithms for generalized fractional programming the purpose of this paper is to analyze the convergence of interval type algorithms for solving the generalized fractional program they are characterized by an interval lb k ub k including and the length of the interval is reduced at each iteration a closer analysis of the bounds lb k and ub k allows to modify slightly the best known interval type algorithm newmodm accordingly to prove its convergence and derive convergence rates similar to those for a dinkelbach type algorithm maxmodm under the same conditions numerical results in the linear case indicate that the modifications to get convergence results are not obtained at the expense of the numerical efficiency since the modified version bfii is as efficient as newmodm and more efficient than maxmodm

ten methods to bound multiple roots of polynomials given a univariate polynomial p with a k fold multiple root or a k fold root cluster near some z we discuss nine different methods to compute a disc near z which either contains exactly or contains at least k roots of p many of the presented methods are known some are new we are especially interested in the behaviour of methods when implemented in a rigorous way that is when taking into account all possible effects of rounding errors in other words every result shall be mathematically correct we display extensive test sets comparing the methods under different circumstances based on the results we present a tenth a hybrid method combining five of the previous methods which for given z i detects the number k of roots near z and ii computes an including disc with in most cases a radius of the order of the numerical sensitivity of the root cluster therefore the resulting discs are numerically nearly optimal

algorithms for generalized fractional programming a generalized fractional programming problem is specified as a nonlinear program where a nonlinear function defined as the maximum over several ratios of functions is to be minimized on a feasible domain of n the purpose of this paper is to outline basic approaches and basic types of algorithms available to deal with this problem and to review their convergence analysis the conclusion includes results and comments on the numerical efficiency of these algorithms

enhanced bisection strategies for the maximin efficiency ratio model abstract the maximin efficiency ratio mer model may be regarded as a new form of data envelopment analysis dea model which is based on the maximum decisional efficiency mde estimation principle the mer model involves a special class of generalized fractional programming problems this paper proposes some improved computational algorithms as alternatives to the original bolzano search and other general purpose algorithms for solving the mer model one of these which may be called a cutting wedges algorithm is similar to the algorithms of the cutting planes type but iteratively modifies coefficients of existing constraints rather than adding new constraints a numerical illustration is presented for the new algorithm a combination of the cutting wedges and the general purpose dinkelbach strategies leads to an enhanced bisection algorithm which improves either the upper or lower bound at all iterations

minimizing the condition number of a gram matrix the condition number of a gram matrix defined by a polynomial basis and a set of points is often used to measure the sensitivity of the least squares polynomial approximation given a polynomial basis we consider the problem of finding a set of points and or weights which minimizes the condition number of the gram matrix the objective function f in the minimization problem is nonconvex and nonsmooth we present an expression of the clarke generalized gradient of f and show that f is clarke regular and strongly semismooth moreover we develop a globally convergent smoothing method to solve the minimization problem by using the exponential smoothing function to illustrate applications of minimizing the condition number we report numerical results for the gram matrix defined by the weighted vandermonde like matrix for least squares approximation on an interval and for the gram matrix defined by an orthonormal set of real spherical harmonics for least squares approximation on the sphere

duality in generalized linear fractional programming we consider a generalization of a linear fractional program where the maximum of finitely many linear ratios is to be minimized subject to linear constraints for this min max problem a dual in the form of a max min problem is introduced and duality relations are established

a new algorithm for generalized fractional programs a new dual problem for convex generalized fractional programs with no duality gap is presented and it is shown how this dual problem can be efficiently solved using a parametric approach the resulting algorithm can be seen as dual to the dinkelbach type algorithm for generalized fractional programs since it approximates the optimal objective value of the dual primal problem from below convergence results for this algorithm are derived and an easy condition to achieve superlinear convergence is also established moreover under some additional assumptions the algorithm also recovers at the same time an optimal solution of the primal problem we also consider a variant of this new algorithm based on scaling the dual parametric function the numerical results in case of quadratic linear ratios and linear constraints show that the performance of the new algorithm and its scaled version is superior to that of the dinkelbach type algorithms from the computational results it also appears that contrary to the primal approach the dual approach is less influenced by scaling

a fast approach to deformable surface 3d tracking deformable surface 3d tracking is a severely under constrained problem and great efforts have been made to solve it a recent state of the art approach solves this problem by formulating it as a second order cone programming socp problem however one drawback of this approach is that it is time consuming in this paper we propose an effective method for 3d deformable surface tracking first we formulate the deformable surface tracking problem as a linear programming lp problem then we solve the lp problem with an algorithm which converges superlinearly rather than bisection algorithm whose convergence speed is linear our experimental studies on synthetic and real data have demonstrated the proposed method can not only reliably recover 3d structures of surfaces but also run faster than the state of the art method

on the posynomial fractional programming problems abstract the posynomial fractional programming pfp problem arises from the summation minimization of several quotient terms which are composed of posynomial terms appearing in the objective function subject to given posynomial constraints this paper proposes an approximate approach to solving a pfp problem a linear programming relaxation is derived for the problem based on piecewise linearization techniques which first convert a posynomial term into the sum of absolute terms these absolute terms are then linearized by some linearization techniques the proposed approach could reach a solution as close as possible to a global optimum

parametric approaches to fractional programs the fractional program p is defined by maxf x g x subject tox x a class of methods for solving p is based on the auxiliary problem q with a parameter maxf x g x subject tox x starting with two classical methods in this class the newton method and the binary search method a number of variations are introduced and compared among the proposed methods the modified binary search method is theoretically interesting because of its superlinear convergence and the capability to provide an explicit interval containing the optimum parameter value bar lambda computational behavior is tested by solving fractional knapsack problems and quadratic fractional programs the interpolated binary search method seems to be most efficient while other methods also behave surprisingly well

fractional qcqp with applications in ml steering direction estimation for radar detection this paper deals with the problem of estimating the steering direction of a signal embedded in gaussian disturbance under a general quadratic inequality constraint representing the uncertainty region of the steering we resort to the maximum likelihood ml criterion and focus on two scenarios the former assumes that the complex amplitude of the useful signal component fluctuates from snapshot to snapshot the latter supposes that the useful signal keeps a constant amplitude within all the snapshots we prove that the ml criterion leads in both cases to a fractional quadratically constrained quadratic problem qcqp in order to solve it we first relax the problem into a constrained fractional semidefinite programming sdp problem which is shown equivalent via the charnes cooper transformation to an sdp problem then exploiting a suitable rank one decomposition we show that the sdp relaxation is tight and give a procedure to construct in polynomial time an optimal solution of the original problem from an optimal solution of the fractional sdp we also assess the quality of the derived estimator through a comparison between its performance and the constrained cramer rao lower bound crb finally we give two applications of the proposed theoretical framework in the context of radar detection

globally optimal energy efficient power control and receiver design in wireless networks the characterization of the global maximum of energy efficiency ee problems in wireless networks is a challenging problem due to their nonconvex nature in interference channels the aim of this paper is to develop a new and general framework to achieve globally optimal solutions first the hidden monotonic structure of the most common ee maximization problems is exploited jointly with fractional programming theory to obtain globally optimal solutions with exponential complexity in the number of network links to overcome the high complexity we also propose a framework to compute suboptimal power control strategies with affordable complexity this is achieved by merging fractional programming and sequential optimization the proposed monotonic framework is used to shed light on the ultimate performance of wireless networks in terms of ee and also to benchmark the performance of the lower complexity framework based on sequential programming numerical evidence is provided to show that the sequential fractional programming framework achieves global optimality in several practical communication scenarios

critical behavior at the theta point of self avoiding walks on a manhattan lattice the authors study the collapse of self attracting self avoiding walks on a manhattan lattice by means of phenomenological renormalization group techniques their results support a recent prediction of nonuniversality for the entropic exponent gamma the surface exponents are in the same universality class as the theta transition on undirected lattices thus differing from those of the theta point they use a two parameter renormalization group to obtain estimates of the crossover exponent

a parametric method for solving the linear fractional programming problem this paper describes an approach for determining the optimal solution of the linear fractional programming problem this approach is based mainly on a parametric analysis of a related linear substitution problem whereas existing algorithms for linear fractional programs concentrate solely on determining the optimal solution the approach described in this paper provides the decision maker with additional insights and valuable information on the underlying decision problem when the usual algorithms are applied this additional information which is gained by the parametric analysis of the substitution problem can be obtained only by further computational effort

an algorithm for a class of nonconvex programming problems with nonlinear fractional objectives in public policy decision making and in capital planning fractional criterion functions occur for a given set of desirable target values goals i this paper develops an algorithm for solving a nonconvex programming problem of the type minx s maxi i fi x gi x i i 1 m where fi are convex functions gi are concave functions over the convex subset s of rn and i are nondecreasing gauge functions here i is the penalty incurred whenever the fractional objective fi gi deviates from the target value i the problem is then to choose an x that minimizes the maximum penalty incurred

coordinated beamforming for energy efficient transmission in multicell multiuser systems in this paper we study energy efficient joint power allocation and beamforming for coordinated multicell multiuser downlink systems the considered optimization problem is in a non convex fractional form and hard to tackle we propose to first transform the original problem into an equivalent optimization problem in a parametric subtractive form by which we reach its solution through a two layer optimization scheme the outer layer only involves one dimension search for the energy efficiency parameter which can be addressed using the bi section search the key issue lies in the inner layer where a non fractional sub problem needs to tackle by exploiting the relationship between the user rate and the mean square error we then develop an iterative algorithm to solve it the convergence of this algorithm is proved and the solution is further derived in closed form our analysis also shows that the proposed algorithm can be implemented in parallel with reasonable complexity numerical results illustrate that our algorithm has a fast convergence and achieves near optimal energy efficiency it is also observed that at the low transmit power region our solution almost achieves the optimal sum rate and the optimal energy efficiency simultaneously while at the middle high transmit power region a certain sum rate loss is suffered in order to guarantee the energy efficiency

a fast algorithm for a class of generalized fractional programs in many decision problems criteria occur that can be expressed as ratios the corresponding optimization problems are nonconvex programs of fractional type in this paper an algorithm for the numerical solution of these problems is introduced that converges always at superlinear speed numerical examples are presented

bounds for eigenvalues of matrix polynomials upper and lower bounds are derived for the absolute values of the eigenvalues of a matrix polynomial or matrix the bounds are based on norms of the coefficient matrices and involve the inverses of the leading and trailing coefficient matrices they generalize various existing bounds for scalar polynomials and single matrices a variety of tools are used in the derivations including block companion matrices gershgorin s theorem the numerical radius and associated scalar polynomials numerical experiments show that the bounds can be surprisingly sharp on practical problems

fractional programming revisited abstract a new format is proposed for fractional programming problems this format gives full expression to the fact that the parametric approach to fractional programming problems is rooted in a first order necessary and sufficient optimality condition it is thus shown that although traditionally it has not been construed as such the parametric approach is in fact classical par excellence

generalized cheney loeb dinkelbach type algorithms we present a class of generalized cheney loeb dinkelbach type i e dc type algorithms together with the problems they solve and discuss convergence properties our algorithms which are as simple as the dc one are applicable to classes of programs larger than fractional in two particular cases the first being an essentially max min problem and the second a pareto extremum we prove 1 61 and second order convergence respectively applications are given to extended fractional max min and fractional pareto extremum problems

linear programming system identification we define a version of the inverse linear programming problem that we call linear programming system identification this version of the problem seeks to identify both the objective function coefficient vector and the constraint matrix of a linear programming problem that best fits a set of observed vector pairs one vector is that of actual decisions that we call outputs these are regarded as approximations of optimal decision vectors the other vector consists of the inputs or resources actually used to produce the corresponding outputs we propose an algorithm for approximating the maximum likelihood solution the major limitation of the method is the computation of exact volumes of convex polytopes a numerical illustration is given for simulated data

generalized fractional programming algorithms and numerical experimentation abstract several algorithms to solve the generalized fractional program are summarized and compared numerically in the linear case these algorithms are iterative procedures requiring the solution of a linear programming problem at each iteration in the linear case the most efficient algorithm is obtained by marrying the newton approach within the dinkelbach approach for fractional programming