Constrained global quasi concave minimization problems are known to be nphard in general. Let us observe that monotone functions in r 1 are both quasi concave and quasi convex, because both the upperand the lower level sets are convex they are intervals. We point out that various other classes of global optimization problems can be expressed in this way. In your example, fx is monotone increasing up to f0, and then monotone decreasing after. Differentiable quasiconvex function tutorialspoint. Quasi concave programming 781 function can always be so extended.
Jul, 2006 a new method for maximizing a concave quadratic function with bounds on the variables is introduced. Nonlinear programming studies the general case in which the objective function or the constraints or both contain nonlinear parts. Quasi concavity and maximization quasi concave functions have nice properties for maximization. Quasiconvex optimization and location theory applied. This shows that the sum of two quasi convex functions is not necessarily quasi convex.
As you can verify, it also saqsfies the condition for nonstrict quasi convexity, but fails the condition. Concave programming 1 introduction concave programming is another special case of the general constrained optimization problem max x x subject to gx. Math 20070 optimization in finance quasiconcave and quasi convex functions prove that the following functions are. Interactive evolutionary multiobjective optimization for. Siam journal on applied mathematics society for industrial. I if f is concave, then it is quasiconcave, so you might start by checking for concavity. Quasi concave and convex best online classes for eco. We present two algorithms, which can be seen as slight modifications of bensontype algorithms for multiple objective linear programs. Pdf preference elicitation and robust optimization with. The new algorithm combines conjugate gradients with gradient projection techniques, as the algorithm of more and toraldo siam j. A unified approach to inequality and equality constraints.
Lagrange multipliers, using tangency to solve constrained optimization duration. Concave minimization f is concave, d is convex continuous global optimization f is continuous, d is compact differential convex d. It also can be shown that any monotonic nondecreasing function of a quasi concave. Before casting the synthesis in the format presented in this paper, lyapunovbased piecewiseaf. Concave implies logconcave, and it is known that the product of logconcave functions is logconcave, hence quasi. Interactive evolutionary multiobjective optimization for quasiconcave preference functions article in european journal of operational research 2062. Many classes of convex optimization problems admit polynomialtime algorithms, 1 whereas mathematical optimization is in general nphard. May 27, 2019 global optimization problems with a quasi concave objective function and linear constraints are studied.
Importance of quasi concavity and quasi convexity in consumer optimization problems deepinder kaur1, harmeet singh gulati2 1assistant professor, mathematics department, s. The product of two positive concave or logconcave functions is quasiconcave. We propose two approaches based respectively on the support functions and level functions of quasi concave functions to develop tractable formulations of the maximin preference robust optimization model. Stochastic proximal quasi newton methods for nonconvex composite optimization. In this work we investigate a duality between quasi concave set functions and linkage functions. A function fis concave or strictly concave if fis convex or strictly convex, respectively a ne functions, i. For example logx is concave, and it is quasi convex. An fptas for minimizing a class of lowrank quasiconcave. A non concave function isnt a widely used term, and its sufficient to say its a function that isnt concave though ive seen it used to refer to nonconvex functions. Configuration optimization design of ti6al4v lattice. The problem of quasiconvex optimization is to find an f in k so that a iiffll infilfkll. Quasiconcave functions on antimatroids internet archive. The modification of the molp algorithms results into a. Theorem 1 shows that any function, subject to weak conditions, can be expressed as the sum of a convex and concave part this decomposition is not unique.
The established results can be used to provide powerful tools for sketching numerical algorithms and. Obviously, the sum of uniformly quasi concave functions, on the same set, is also quasi concave and if the functions are also nonnegative, then the same holds for their product as well. Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets. Cvx is a matlabbased modeling system for convex optimization. In this paper, we establish necessary and sufficient conditions to characterize weakly efficient solutions in nonsmooth quasiconvex multiobjective programming. We propose two approaches based respectively on the.
View notes quasi from math 2007 at university college dublin. A vector linear programming approach for certain global. Some software does exist for solving quasiconvex problems e. Quasiconcave optimization of secrecy redundancy rate in. More significantly, first order conditions are not in general sufficient to identify global of quasi concave or convex optimization problems. Part of the nonconvex optimization and its applications book series noia, volume 2 the purpose of this chapter is to present the essential elements of the theory, applications, and solution algorithms of concave minimization. On the maximization of a concave quadratic function with. Any convex optimization problem has geometric interpretation. We point out that various other classes of global optimization problems can be expressed in. Global optimization problems with a quasi concave objective function and linear constraints are studied. Concave and convex, quasi concave and quasi convex and economic optimization posted on november 6, 2008 leave a comment one point that i myself seem to igore like most economist student is about concave and convex problem but they are quite important in building economic model of optimization. The latter property is used in the next section, where we prove our main result. If a function is increasing, then is it both quasiconcave. In this paper, we shall usually deal with functions quasi concave over nonnegative values of the variables.
Characterization of the weakly efficient solutions in. The special class of concave fractional programs can be transformed to a convex optimization problem. The robust choice function is increasing and quasiconcave but not necessarily translation invariant, a key property of monetary risk measures. If a given optimization problem can be transformed to a convex equivalent, then this interpretive benefit is acquired. Portfolio optimization models are used throughout the. Karushkuhntucker optimality necessary conditions algorithms for convex problems. A challenging fundamental issue still remains that it is not clear whether the local optimum computed by a given optimization.
This also means that if a monotonic transformation of f is concave, then f is concave. Abstract convergence theorem for quasiconvex optimization. Can you give an example of a quasi concave function that is not concave. In a hybrid automatic repeat request with chase combining harqcc system, we analyze physical layer secure performance and determine the secrecy redundancy rate by proposed quasi concave optimization. These are nonlinear models that are used to determine. A unimodal function has the property that it is monotone increasing up to a point, and then. The 4th conference on optimization methods and software. Their behavior on antimatroids was studied in kempner, muchnik 2003, where they were applied to constraint clustering.
We present two algorithms, which can be seen as slight modifications of bensontype algorithms for multiple objective linear programs molp. In addition to these analytical, high level contributions, we develop cognitive networks for. Matlab software for disciplined convex programming. An interactive evolutionary multiobjective optimization algorithm with a limited number of decision maker calls, european journal of operational research, elsevier, vol. Preference elicitation and robust optimization with multi. On the maximization of a concave quadratic function with box. Weve added some interesting new features for users and system administrators. A local maximum of a quasi concave function need not also be a global maximum of the function. Whats the difference between quasiconcavity and concavity. All journal articles featured in optimization methods and software vol 35 issue 2. A new method for maximizing a concave quadratic function with bounds on the variables is introduced. The results are proved in terms of the greenbergpierskalla, penot, plastria, gutierrez and suzukikuroiwa subdifferentials. Nov 18, 2011 the algorithm is simple and based on a linearprogramming reformulation of the general convexfunction interpolation problem see, for example, p. There do not exist local criteria to characterize a point as a global minimum.
In this paper, we consider the problem of minimizing a lowrank quasi concave. A quasiconvex function is a realvalued function defined on an interval or on a convex subset of a real vector space such that the. In the case of bounded polyhedral constraints it is well known that a quasi concave function attains its minimum in a vertex of the feasible region. A convex optimization problem is a problem where all of the constraints are convex functions, and the objective is a convex function if minimizing, or a concave function if maximizing. Maximum likelihood estimation of a log concave probability density is formulated as a convex optimization problem and shown to have an equivalent dual formulation as a constrained maximum shannon entropy problem. Mathematical optimization alternatively spelt optimisation or mathematical programming is the selection of a best element with regard to some criterion from some set of available alternatives.
Furthermore, we found the quasi concave class has other desirable properties, reacting well to the absence of certain kinds of information and degrading gracefully under reduced network control. In fact, this implies that if a function is convcave then thats also quasiconcave but not necessarily the converse is true. Strong oracle optimality of folded concave penalized. Therefore, concave or quasi concave minimization forms an important class of nonconvex optimization problems that are computationally intractable in general.
Chapter 12 optimization with equality constraints 365 figure 12. In mathematics, computer science, economics, or management science, mathematical optimization alternatively, optimization or mathematical programming is the selection of a best element with regard to some criteria from some set of available alternatives. Uniform quasiconcavity in probabilistic constrained. I if f is a monotonic transformation of a concave function, it is quasiconcave. Optimization problem types convex optimization solver. In this work we investigate a duality between quasi concave. Linear functions are convex, so linear programming problems are convex problems. The robust choice function is increasing and quasi concave but not necessarily translation invariant, a key property of monetary risk measures. What are some examples of quasiconvex functions which are not. Is negative quadratic function quasiconvex stack overflow. A function which is both quasi convex and quasi concave is called quasi linear. A major part of modern optimization theory is concerned with establishing under suitableconditions anequivalenceresult betweena point of constrained global maximum and saddle point.
Concave and convex, quasiconcave and quasiconvex and. A unimodal function has the property that it is monotone increasing up to a point, and then monotone decreasing after that. We propose two approaches based respectively on the support functions and level functions of quasi concave functions to develop tractable formulations of the maximin preference robust optimization. A successive underestimation method for concave minimization. Disciplined quasiconvex programming stanford university. While the objective function for a statistical model is often concave in a neighborhood of the optimal solution, that neighborhood is often a small proportion. Interactive evolutionary multiobjective optimization for quasiconcave preference functions author links open overlay panel john w.
Sep 28, 2018 quasi static uniaxial compression performance experiments and finite element analysis fea are conducted on an afcc structure and the control group bodycentered cubic bcc structure. Quasiconvex and quasiconcave functions tutorialspoint. Its the same function as before, but with different powers. Quasiconvex optimization and location theory applied optimization 9 1998th edition. Genetic optimization using derivatives in r tion that is not globally concave may have multiple local optima, saddle points, boundary solutions or discontinuities. Jul 06, 2016 there are many examples of quasi convex functions that are not convex. A concave function is the negative of a convex function. Optimization is the science of making a best choice in the face of conflicting requirements. Importance of quasiconcavity and quasi convexity in. The results demonstrates that our optimized afcc lattice structure is superior to the bcc structure, with elastic modulus and yield limit increases of 143% and. Stochastic proximal quasinewton methods for nonconvex.
823 845 1328 1336 1365 1361 894 696 761 917 485 230 341 661 327 442 1363 436 773 31 1104 838 1492 863 1016 454 71 424 557 296 332 171 1092 1431 972 430 368 1459 1269 831 435 830 127 1052 1054 1116 605 1235 130