Quasiconvex Objective Function Research Articles

Relaxed-inertial proximal point type algorithms for quasiconvex minimization

We propose a relaxed-inertial proximal point type algorithm for solving optimization problems consisting in minimizing strongly quasiconvex functions whose variables lie in finitely dimensional linear subspaces. A relaxed version of the method where the constraint set is only closed and convex is also discussed, and so is the case of a quasiconvex objective function. Numerical experiments illustrate the theoretical results.

Characterizations of Solution Sets of Differentiable Quasiconvex Programming Problems

In this paper, we study some problems with continuously differentiable quasiconvex objective function. We prove that exactly one of the following two alternatives holds: (I) the gradient of the objective function is different from zero over the solution set and the normalized gradient is constant over it; (II) the gradient of the objective function is equal to zero over the solution set. As a consequence, we obtain characterizations of the solution set of a quasiconvex continuously differentiable program, provided that one of the solutions is known. We also derive Lagrange multiplier characterizations of the solutions set of an inequality constrained problem with continuously differentiable objective function and differentiable constraints, which are all quasiconvex on some convex set, not necessarily open. We compare our results with the previous ones. Several examples are provided.

Energy-Efficient Multi-Stream Carrier Aggregation for Heterogeneous Networks in 5G Wireless Systems

Multi-stream carrier aggregation (MSCA) has been recently proposed as a mechanism to increase the amount of bandwidth available to users for heterogeneous networks (HetNets) in 5G wireless systems. Previous studies have focused only on maximizing the network capacity and fairness, without considering the energy efficiency of the MSCA. In this paper, the use of MSCA to minimize the energy consumption in a multi-layer HetNet is studied. The convexity of the energy minimization problem is examined, leading to the need of a quasi-convex relaxation. With this approximation, an algorithm bisection method for energy minimization is designed to solve the energy minimization problem and obtains an optimum cell-association policy. Since the operators are generally interested in a balance between the energy minimization and capacity maximization, such multi-objective optimization is needed, and we studied it in this paper. The two aforementioned conflicting objectives can be jointly analyzed and solved through scalarization, even though the energy minimization has a quasi-convex objective function, and not a convex one. Performance evaluation is provided to identify the achievable energy savings of our proposed algorithm and to characterize the tradeoffs between the energy minimization and capacity maximization in a multi-layer HetNet in 5G systems that support MSCA.

Slater CQ, optimality and duality for quasiconvex semi-infinite optimization problems

In this paper, we introduce and study the Slater constraint qualification (CQ) for a semi-infinite optimization problem with upper-semicontinuous quasiconvex objective and constraint functions. Then, some Karush–Kuhn–Tucker (KKT) type necessary and sufficient optimality conditions as well as duality results are derived. The final part of the paper is devoted to a linear characterization of optimality and the gap function for considered semi-infinite problem.

An inexact proximal method for quasiconvex minimization

In this paper we propose an inexact proximal point method to solve constrained minimization problems with locally Lipschitz quasiconvex objective functions. Assuming that the function is also bounded from below, lower semicontinuous and using proximal distances, we show that the sequence generated for the method converges to a stationary point of the problem.

Continuity of Optimal Solution Functions and their Conditions on Objective Functions

When an optimization problem has a set of parameters besides decision variables, the mapping from the parameters to the corresponding optimal solutions is determined accordingly. This mapping, either single-valued or set-valued, is called an optimal solution function in the literature of parametric optimization and a generalized inverse mapping in the literature of inverse problems. We present a pair of relationships between classes of objective functions and optimal solution functions in the form of two theorems. The first theorem states that any continuous quasi-convex objective function results in an upper semicontinuous optimal solution function. This is a variant of the Berge maximum theorem, wherein we replace the condition of compact-valuedness of the feasible solution function with other assumptions. The second theorem states that any optimal solution function with a certain continuity can be realized by a continuous convex objective function if the feasible solutions form an affine subspace. This...

ON SURROGATE DUALITY FOR ROBUST SEMI-INFINITE OPTIMIZATION PROBLEM

Abstract. A semi-inflnite optimization problem involving a quasi-convex objective function and inflnitely many convex constraintfunctions with data uncertainty is considered. A surrogate dualitytheorem for the semi-inflnite optimization problem is given undera closed and convex cone constraint qualiflcation. 1. IntroductionOptimization problems in the face of data uncertainty have beentreated by the worst case approach(the robust approach) or the sto-chastic approach. The worst case approach for optimization problems,which has emerged as a powerful deterministic approach for studyingoptimization problems with data uncertainty, associates an uncertainoptimization problem with its robust counterpart. Many researchers[1, 6, 7, 12] have investigated duality theory for linear or convex pro-gramming problems under uncertainty with the worst case approach.On the other hand, recently, many authors [3, 4, 8, 9, 10, 11, 12]investigated surrogate duality for quasiconvex programming. Surrogateduality is used in not only quasi-convex programming but also integerprogramming and the knapsack problem [2, 3, 4, 8, 9, 10]. In particualr,Suzuki, Kuroiwa and Lee [12] proved a surrogate duality theorem for anoptimization problem involving a quasi-convex objective function andflnitely many convex constraint functions with data uncertainty, and a

A proximal method with separable Bregman distances for quasiconvex minimization over the nonnegative orthant

We present an interior proximal method with Bregman distance, for solving the minimization problem with quasiconvex objective function under nonnegative constraints. The Bregman function is considered separable and zone coercive, and the zone is the interior of the positive orthant. Under the assumption that the solution set is nonempty and the objective function is continuously differentiable, we establish the well definedness of the sequence generated by our algorithm and obtain two important convergence results, and show in the main one that the sequence converges to a solution point of the problem when the regularization parameters go to zero.

Asymptotic convergence of an inertial proximal method for unconstrained quasiconvex minimization

This paper deals with the convergence analysis of a second order proximal method for approaching critical points of a smooth and quasiconvex objective function defined on a real Hilbert space. The considered method, well-known in the convex case, unifies proximal method, relaxation and inertial-type extrapolation. The convergence theorems established in this new setting improve recent ones.

The gradient and heavy ball with friction dynamical systems: the quasiconvex case

We consider the gradient system $$\dot x(t)+\nabla\phi(x(t))=0$$ and the so-called heavy ball with friction dynamical system $$\ddot x(t) +\lambda\dot x(t)+\nabla\phi(x(t))=0$$, as well as an implicit discrete (proximal) version of it, and study the asymptotic behavior of their solutions in the case of a smooth and quasiconvex objective function Φ. Minimization properties of trajectories are obtained under various additional assumptions. We finally show a minimizing property of the heavy ball method which is not shared by the gradient method: the genericity of the convergence of each trajectory, at least when Φ is a Morse function, towards local minimum of Φ.

Optimality Conditions for Quasiconvex Programs

We present necessary and sufficient optimality conditions for a problem with a convex set constraint and a quasiconvex objective function. We apply the obtained results to a mathematical programming problem involving quasiconvex functions.

Structure of efficient sets in lexicographic quasiconvex multicriteria optimization

The aim of this paper is to emphasize some remarkable properties of multicriteria optimization problems involving lexicographic quasiconvex objective functions. It is shown that, under appropriate assumptions, these problems are Pareto reducible and their efficient sets are strongly contractible.

Ringing reduction in image restoration algorithms using a constraint on the inferior bound of the solution

For many image reconstruction (deconvolution) problems, particularly in the field of astrophysics, the “ringing” effect around bright point sources (or more generally in the vicinity of abrupt variations of the intensity) is a prominent artifact. This effect appears when point sources are superimposed on a background and when the reconstruction algorithm do not take into account such background. For astronomical images, difficulties due to this artifact arise during photometric measurements of point sources or in estimating low intensity structures around point sources. We propose a general method allowing to devise maximum-likelihood image restoration algorithms with a general constraint on the inferior bound of the solution, this bound can be different at each point of the signal; ringing phenomena are then reduced. The method is founded on the use of the Kuhn–Tucker first order optimality conditions and can be applied to the minimization of any convex objective function whose definition domain contains the domain of the constraints. We consider Gaussian and Poisson noise processes leading to the minimization of convex objective functions. Moreover, the method is extended to Gamma likelihood as proposed recently by Cao et al. (IEEE Trans. Image, Process. 8(2) (1999) 286). This correspond to the minimization of a quasi-convex objective function: the Itakura–Saito distance. In each case, we develop non-relaxed algorithms, written in a “quasi multiplicative” form, very easy to use, and whose convergence is formally demonstrated in the Gaussian case. The classical EM and ISRA algorithms as well as those by Cao et al. (IEEE Trans. Image, Process. 8(2) (1999) 286) appear as particular cases of the proposed algorithms. The relaxed and accelerated forms of the algorithms are given. In that case, the convergence is ensured, and we show that “economic methods” for the stepsize computation, allow to obtain fast algorithms. This permits to avoid time expensive computations of the “precision” line search methods generally used. Accelerated forms deduced from the method proposed in (Signal processing 81(5) (2001) 945) are exhibited. The algorithms are tested on simulated images and applied to real cases. The properties of the algorithms are shown, in particular their ability to suppress the “ringing” phenomena.

Convergence of Implementable Descent Algorithms for Unconstrained Optimization

Descent algorithms use sufficient descent directions combined with stepsize rules, such as the Armijo rule, to produce sequences of iterates whose cluster points satisfy some necessary optimality conditions. In this note, we present a proof that the whole sequence of iterates converges for quasiconvex objective functions.

Multiple local minima in radiotherapy optimization problems with dose-volume constraints.

The cause of multiple local minima in beam weight optimization problems subject to dose-volume constraints is analyzed. Three objective functions were considered: (a) maximization of tumor control probability (TCP), (b) maximization of the minimum target dose, and (c) minimization of the mean-squared-deviation of the target dose from the prescription dose. It is shown that: (a) TCP models generally result in strongly quasiconvex objective functions; (b) maximization of the minimum target dose results in a strongly quasiconvex objective function; and (c) minimizing the root-mean-square dose deviation results in a convex objective function. Dose-volume constraints are considered such that, for each region at risk (RAR), the volume of tissue whose dose exceeds a certain tolerance dose (DTol) is kept equal to or below a given fractional level (VTol). If all RARs lack a "volume effect" (i.e., VTol = 0 for all RARs) then there is a single local minimum. But if volume effects are present, then the feasible space is possibly nonconvex and therefore possibly leads to multiple local minima. These conclusions hold for all three objective functions. Hence, possible local minima come not from the nonlinear nature of the objective functions considered, but from the "either this volume or that volume but not both" nature of the volume effect. These observations imply that optimization algorithms for dose-volume constraint types of problems should have effective strategies for dealing with multiple local minima.

On sufficient optimality conditions for a quasiconvex programming problem

Some remarks are made on a paper by Bector, Chandra, and Bector (see Ref. 1) concerning the Fritz John and Kuhn-Tucker sufficient optmality conditions as well as duality theorems for a nonlinear programming problem with a quasiconvex objective function.

Technical Note—On Min-Max Integer Allocation Problems

This note treats a bottleneck integer allocation problem. The results obtained supercede those in this Journal due to Zeitlin (Zeitlin, Z. 1981. Integer allocation problems of min-max type with quasiconvex separable functions. Opns. Res. 29 207–211.), who has considered a similar allocation problem with a strictly quasiconvex objective function.

Mathematical programming problems with quasi-convex objective functions

This paper generalizes results obtained by Arrow and Enthoven in [1] for functions of ClassC2 on the non-negative orthant.

Quadratic Programming with a Quasiconvex Objective Function

This paper gives both necessary and sufficient conditions for a quadratic function to be quasiconvex in the nonnegative orthant. Methods of pseudoconvex programming (such as those of Frank and Wolfe) can solve linearly constrained quadratic programming problems with such an objective function.