Showing papers on "Concave function published in 2005"

Geometry of Log-concave Functions and Measures

Abstract: We present a view of log-concave measures, which enables one to build an isomorphic theory for high dimensional log-concave measures, analogous to the corresponding theory for convex bodies. Concepts such as duality and the Minkowski sum are described for log-concave functions. In this context, we interpret the Brunn–Minkowski and the Blaschke–Santalo inequalities and prove the two corresponding reverse inequalities. We also prove an analog of Milman’s quotient of subspace theorem, and present a functional version of the Urysohn inequality.

A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs

Abstract: We propose a branch-and-bound algorithm for solving nonconvex quadratically-constrained quadratic programs. The algorithm is novel in that branching is done by partitioning the feasible region into the Cartesian product of two-dimensional triangles and rectangles. Explicit formulae for the convex and concave envelopes of bilinear functions over triangles and rectangles are derived and shown to be second-order cone representable. The usefulness of these new relaxations is demonstrated both theoretically and computationally.

A Representation Result for Concave Schur Concave Functions

Abstract: A representation result is provided for concave Schur concave functions on L proportional to infinity (Omega). In particular, it is proven that any monotone concave Schur concave weakly upper semicontinuous function is the infinimum of a family of nonnegative affine combinations of Choquet integrals with respect to a convex continuous distortion of the underlying probability. The method of proof is based on the concave Fenchel transform and on Hardy and Littlewood's inequality. Under the assumption that the probability space is nonatomic, concave, weakly upper semicontinuous, law-invariant functions are shown to coincide with weakly upper semicontinuous concave Schur concave functions. A representation result is, thus, obtained for weakly upper semicontinuous concave law-invariant functions.

A representation result for concave schur concave functions

Abstract: A representation result is provided for concave Schur concave functions on L∞(Ω). In particular, it is proven that any monotone concave Schur concave weakly upper semicontinuous function is the infinimum of a family of nonnegative affine combinations of Choquet integrals with respect to a convex continuous distortion of the underlying probability. The method of proof is based on the concave Fenchel transform and on Hardy and Littlewood's inequality. Under the assumption that the probability space is nonatomic, concave, weakly upper semicontinuous, law-invariant functions are shown to coincide with weakly upper semicontinuous concave Schur concave functions. A representation result is, thus, obtained for weakly upper semicontinuous concave law-invariant functions.

Integrated Lot Sizing in Serial Supply Chains with Production Capacities

Abstract: We consider a model for a serial supply chain in which production, inventory, and transportation decisions are integrated in the presence of production capacities and concave cost functions. The model we study generalizes the uncapacitated serial single-item multilevel economic lot-sizing model by adding stationary production capacities at the manufacturer level. We present algorithms with a running time that is polynomial in the planning horizon when all cost functions are concave. In addition, we consider different transportation and inventory holding cost structures that yield improved running times: inventory holding cost functions that are linear and transportation cost functions that are either linear, or are concave with a fixed-charge structure. In the latter case, we make the additional common and reasonable assumption that the variable transportation and inventory costs are such that holding inventories at higher levels in the supply chain is more attractive from a variable cost perspective. While the running times of the algorithms are exponential in the number of levels in the supply chain in the general concave cost case, the running times are remarkably insensitive to the number of levels for the other two cost structures.

Network Utility Maximization With Nonconcave Utilities Using Sum-of-Squares Method

Abstract: The Network Utility Maximization problem has recently been used extensively to analyze and design distributed rate allocation in networks such as the Internet. A major limitation in the state-of-the-art is that user utility functions are assumed to be strictly concave functions, modeling elastic flows. Many applications require inelastic flow models where nonconcave utility functions need to be maximized. It has been an open problem to find the globally optimal rate allocation that solves nonconcave network utility maximization, which is a difficult nonconvex optimization problem. We provide a centralized algorithm for off-line analysis and establishment of a performance benchmark for nonconcave utility maximization. Based on the semialgebraic approach to polynomial optimization, we employ convex sum-of-squares relaxations solved by a sequence of semidefinite programs, to obtain increasingly tighter upper bounds on total achievable utility for polynomial utilities. Surprisingly, in all our experiments, a very low order and often a minimal order relaxation yields not just a bound on attainable network utility, but the globally maximized network utility. When the bound is exact, which can be proved using a sufficient test, we can also recover a globally optimal rate allocation. In addition to polynomial utilities, sigmoidal utilities can be transformed into polynomials and are handled. Furthermore, using two alternative representation theorems for positive polynomials, we present price interpretations in economics terms for these relaxations, extending the classical interpretation of independent congestion pricing on each link to pricing for the simultaneous usage of multiple links.

Supply function equilibrium in electricity spot markets with contracts and price caps

Abstract: In electricity wholesale markets, generators often sign long term contracts with purchasers of power in order to hedge risks. In this paper, we consider a market where demand is uncertain, but can be represented as a function of price together with a random shock. Each generator offers a smooth supply function into the market and wishes to maximize his expected profit, allowing for his contract position. We investigate supply function equilibria in this setting, using a model introduced by Anderson and Philpott. We study first the existence of a unique monotonically increasing supply curve that maximizes the objective function under the constraint of limited generation capacity and a price cap, and discuss the influence of the generator’s contract on the optimal supply curve. We then investigate the existence of a symmetric Nash supply function equilibrium, where we do not have to assume that the demand is a concave function of price. Finally, we identify the Nash supply function equilibrium which gives rise to the generators’ maximal expected profit.

Subadditivity of eigenvalue sums

Abstract: Let f(t) be a nonnegative concave function on 0 < t < ∞ with f(0) = 0, and let X, Y be n×n matrices. Then it is known that ||f(|X+Y|)|| 1 ≤ ||f(|X|)|| 1 + ||f(|Y|)|| 1 , where ||·|| 1 is the trace norm. We extend this result to all unitarily invariant norms and prove some inequalities of eigenvalue sums.

Typical convex program is very well posed

Abstract: In this paper we consider the collection of convex programming problems with inequality and equality constraints, in which every problem of the collection is obtained by linear perturbations of the cost function and right-hand side perturbation of the constraints, while the ``core'' cost function and the left-hand side constraint functions are kept fixed. The main result shows that the set of the problems which are not well-posed is σ-porous in a certain strong sense. Our results concern both the infinite and finite dimensional case. In the last case the conclusions are significantly sharper.

A simplicial branch-and-bound algorithm for production-transportation problems with inseparable concave production cost

Abstract: In this paper, we develop a branch-and-bound algorithm to solve a network flow problem of optimizing production and transportation simultaneously. The production cost is assumed to be a concave function in light of scale economy. The proposed algorithm generates a globally optimal solution to this nonconvex minimization problem in finite time, without assuming the separability of the production-cost function unlike existing algorithms. We also report some computational results, which indicate that the algorithm is fairly promising for practical use.

Correlation inequalities: a useful tool in the theory of LDPC codes

Abstract: It is shown that a correlation inequality of statistical mechanics can be applied to low-density parity-check codes. Thanks to this tool we prove that the growth rate of regular LDPC codes, can be exactly calculated by iterative methods, at least on the interval where it is a concave function of the relative weight of code words. We also consider communication over a binary input additive white Gaussian noise channel with a Poisson LDPC code and prove that (at least part of) the GEXIT curve (associated to MAP decoding) can also be computed exactly by the belief propagation decoder. In both problems, the correlation inequality yields sharp lower bounds. We also use a non trivial extension of the interpolation techniques that have recently led to rigorous results in spin glass theory and in the SAT problem

Optimization for products of concave functions

Abstract: Ifh denotes the product of finitely many concave non-negative functions on a compact interval [a, b], then it is shown that there exist pointsα andβ witha≤α≤β≤b such thath is strictly increasing on [α, α), constant on (α, β), and strictly decreasing on (β, b]. This structure theorem leads to an extension of several classical optimization results for concave functions on convex sets to the case of products of concave functions.

Further results related to a minimax problem of Ricceri

Abstract: We deal with a theoric question raised in connection with the application of a three-critical points theorem, obtained by Ricceri, which has been already applied to obtain multiplicity results for boundary value problems in several recent papers. In the settings of the mentioned theorem, the typical assumption is that the following minimax inequality has to be satisfied by some continuous and concave function . When , we have already proved, in a precedent paper, that the problem of finding such function is equivalent to looking for a linear one. Here, we consider the question for any interval and prove that the same conclusion holds. It is worth noticing that our main result implicitly gives the most general conditions under which the minimax inequality occurs for some linear function. We finally want to stress out that although we employ some ideas similar to the ones developed for the case where , a key technical lemma needs different methods to be proved, since the approach used for that particular case does not work for upper-bounded intervals.

Numerical investigation of generalized quasilinearization method for reaction diffusion systems

Abstract: In, we have developed generalized quasilinearization method for reaction diffusion systems when the forcing functions are the sum of convex and concave functions. The solutions of the corresponding linear systems converge monotonically, uniformly and quadratically to the unique solution of the nonlinear problem. As a byproduct of our result, we have discussed the reaction diffusion system when the forcing function satisfies mixed quasimonotone property. In this paper, we have established the application of the theoretical results developed in with numerical examples. We also demonstrate the consistency of the finite different scheme and discuss the stability and convergence of the scheme for the examples considered here.

Convex programming methods for global optimization

Abstract: We describe four approaches to solving nonconvex global optimization problems by convex nonlinear programming methods. It is assumed that the problem becomes convex when selected variables are fixed. The selected variables must be discrete, or else discretized if they are continuous. We first survey some existing methods: disjunctive programming with convex relaxations, logic-based outer approximation, and logic-based Benders decomposition. We then introduce a branch-and-bound method with convex quasi-relaxations (BBCQ) that can be effective when the discrete variables take a large number of real values. The BBCQ method generalizes work of Bollapragada, Ghattas and Hooker on structural design problems. It applies when the constraint functions are concave in the discrete variables and have a weak homogeneity property in the continuous variables.

A method in demand analysis connected with the Monge—Kantorovich problem

Abstract: A method in demand analysis based on the Monge—Kantorovich duality is developed. We characterize (insatiate) demand functions that are rationalized, in different meanings, by concave utility functions with some additional properties such as upper semi-continuity, continuity, non-decrease, strict concavity, positive homogeneity and so on. The characterizations are some kinds of abstract cyclic monotonicity strengthening revealed preference axioms, and also they may be considered as an extension of the Afriat—Varian theory to an arbitrary (infinite) set of ‘observed data’. Particular attention is paid to the case of smooth functions.

Intersection properties for cones of nondecreasing concave functions

Abstract: We prove that the basic facts of the real interpolation method remain true for couples of cones obtained by intersection of the cone of concave functions with rearrangement invariant spaces.

The logarithmic negativity is a full entanglement monotone under measuring LOCC and PPT-operations

Abstract: It is proven that the logarithmic negativity does not increase on average under positive partial transpose preserving (PPT) operation including subselection (a set of operations that incorporate local operations and classical communication (LOCC) as a subset) and, in the process, a further proof is provided that the negativity does not increase on average under the same set of operations. Given that the logarithmic negativity is obtained from the negativity applying a concave function and is itself not a convex quantity this result is surprising as convexity is generally considered as describing the local physical process of losing information. The role of convexity and in particular its relation (or lack thereof) to physical processes is discussed in this context.

Equilibria of Pairs of Nonlinear Maps Associated with Cones

Abstract: Let K1, K2 be closed, full, pointed convex cones in finite-dimensional real vector spaces of the same dimension, and let F : K1 → span K2 be a homogeneous, continuous, K2-convex map that satisfies F(∂K1) ∩ int K2=∅ and FK1 ∩ int K2 ≠ ∅. Using an equivalent formulation of the Borsuk-Ulam theorem in algebraic topology, we show that we have \(F(K_1 \setminus\{0\}) \cap (-K_2)=\emptyset\) and \(K_2 \subseteq FK_1.\) We also prove that if, in addition, G : K1 → span K2 is any homogeneous, continuous map which is (K1, K2)-positive and K2-concave, then there exist a unique real scalar ω0 and a (up to scalar multiples) unique nonzero vector x0 ∈ K1 such that Gx0 = ω0Fx0, and moreover we have ω0 > 0 and x0 ∈ int K1 and we also have a characterization of the scalar ω0. Then, we reformulate the above result in the setting when K1 is replaced by a compact convex set and recapture a classical result of Ky Fan on the equilibrium value of a finite system of convex and concave functions.

A hybrid dynamic programming method for concave resource allocation problems

Abstract: Concave resource allocation problem is an integer programming problem of minimizing a nonincreasing concave function subject to a convex nondecreasing constraint and bounded integer variables This class of problems are encountered in optimization models involving economies of scale In this paper, a new hybrid dynamic programming method was proposed for solving concave resource allocation problems A convex underestimating function was used to approximate the objective function and the resulting convex subproblem was solved with dynamic programming technique after transforming it into a 0–1 linear knapsack problem To ensure the convergence, monotonicity and domain cut technique was employed to remove certain integer boxes and partition the revised domain into a union of integer boxes Computational results were given to show the efficiency of the algorithm

Open shop unit-time scheduling problems with symmetric objective functions

Abstract: The paper addresses the open-shop scheduling problem with unit-time operations and nondecreasing symmetric objective function depending on job completion times. We construct two schedules, one being optimal for any symmetric convex function, the other one for any symmetric concave function. Both schedules are given by analytically defined formulas that determine in O(1) time for each operation the unit-length time slot for its processing.

An Important Property and Application of Geometric Concave Functions

Abstract: If a0, function f:[a,b)→(0,+∞) is a geometric concave function on (a,b), then F(x)=∫ x_af(t)dtis a geometric concave function on (a, b) also and some application of the result be gived.

Nonexistence of solutions in $(0,1)$ for K-P-P-type equations for all $d\ge 1$

Abstract: Consider the KPP-type equation of the form $\Delta u+f(u)=0$, where $f:[0,1] \to \mathbb R_{+}$ is a concave function. We prove for arbitrary dimensions that there is no solution bounded in $(0,1)$. The significance of this result from the point of view of probability theory is also discussed.