Convex Case Research Articles

On Lipschitz Maps, Martingales, and the Radon–Nikodým Property for $${\mathsf{F}}$$ F -Spaces

The development of the Radon–Nikodým property in Banach spaces sprang from the study of the differentiability of Lipschitz maps from the real line into a space X. Even though there was no initial reason to restrict attention to the locally convex case, the absence of genuine convexity set p-Banach spaces apart and the theory evolved without them. Kalton restored the dignity to quasi-Banach spaces and in 1979 he initiated the search for an analogue of the Radon–Nikodým property for these spaces. Continuing in this spirit, we investigate in this setting the connection between differentiability of Lipschitz maps, limits of martingales, and spaces of bounded operators defined on L p for p < 1.

A nonsmooth approach to envelope theorems

We develop a nonsmooth approach to envelope theorems applicable to a broad class of parameterized constrained nonlinear optimization problems that arise typically in economic applications with nonconvexities and/or nonsmooth objectives. Our methods emphasize the role of the Strict Mangasarian–Fromovitz Constraint Qualification (SMFCQ), and include envelope theorems for both the convex and nonconvex case, allow for noninterior solutions as well as equality and inequality constraints. We give new sufficient conditions for the value function to be directionally differentiable, as well as continuously differentiable. We apply our results to stochastic growth models with Markov shocks and constrained lattice programming problems.

Outer Limit of Subdifferentials and Calmness Moduli in Linear and Nonlinear Programming

With a common background and motivation, the main contributions of this paper are developed in two different directions. Firstly, we are concerned with functions, which are the maximum of a finite amount of continuously differentiable functions of n real variables, paying special attention to the case of polyhedral functions. For these max-functions, we obtain some results about outer limits of subdifferentials, which are applied to derive an upper bound for the calmness modulus of nonlinear systems. When confined to the convex case, in addition, a lower bound on this modulus is also obtained. Secondly, by means of a Karush---Kuhn---Tucker index set approach, we are also able to provide a point-based formula for the calmness modulus of the argmin mapping of linear programming problems, without any uniqueness assumption on the optimal set. This formula still provides a lower bound in linear semi-infinite programming. Illustrative examples are given.

On Lipschitz optimization based on gray-box piecewise linearization

We address the problem of minimizing objectives from the class of piecewise differentiable functions whose nonsmoothness can be encapsulated in the absolute value function. They possess local piecewise linear approximations with a discrepancy that can be bounded by a quadratic proximal term. This overestimating local model is continuous but generally nonconvex. It can be generated in its abs-normal form by a minor extension of standard algorithmic differentiation tools. Here we demonstrate how the local model can be minimized by a bundle-type method, which benefits from the availability of additional gray-box information via the abs-normal form. In the convex case our algorithm realizes the consistent steepest descent trajectory for which finite convergence was established earlier, specifically covering counterexamples where steepest descent with exact line-search famously fails. The analysis of the abs-normal representation and the design of the optimization algorithm are geared toward the general case, whereas the convergence proof so far covers only the convex case.

On the convergence of the Sakawa-Shindo algorithm in stochastic control

We analyze an algorithm for solving stochastic control problems, based on Pontrya-gin's maximum principle, due to Sakawa and Shindo in the deterministic case and extended to the stochastic setting by Mazliak. We assume that either the volatility is an affine function of the state, or the dynamics are linear. We obtain a monotone decrease of the cost functions as well as, in the convex case, the fact that the sequence of controls is minimizing, and converges to an optimal solution if it is bounded.

Affine Invariant Distance Using Multiscale Analysis

In this paper we introduce an affine invariant distance definition from a $$2D$$2D point to the boundary of a bounded shape using morphological multiscale analysis. We study the mathematical behavior of this distance by examining separately the cases of convex and non-convex shapes. We prove that the proposed distance is bounded in the convex hull of the shape and infinite otherwise. A numerical scheme is given as well as experiments illustrating the behavior of the affine invariant distance.

A new scalarization method for finding the efficient frontier in non-convex multi-objective problems

One of the most important issues in multi-objective optimization problems (MOPs) is finding Pareto optimal points on the Pareto frontier. This topic is one of the oldest challenges in science and engineering. Many important problems in engineering need to solve a non-convex multi-objective optimization problem (NMOP) in order to achieve the proper results. Gradient based methods, such as Normal Boundary Intersection (NBI), for solving a MOP require solving at least one optimization problem for each solution point. This method can be computationally expensive with an increase in the number of variables and/or constraints of the optimization problem. Nevertheless, the NBI method is a technique motivated by geometrical intuition to provide a better parameterization of the Pareto set than that provided by other techniques. This parameterization is better in the sense that the points obtained by using the NBI method produce a more even coverage of the Pareto curve and this coverage does not miss the interesting middle part of the Pareto curve.This useful property, provides an incentive to create a new method. The first step in this study is using a modified convex hull of individual minimum (mCHIM) in each iteration. The second step is introducing an efficient scalarization problem in order to find the Pareto points on the Pareto front. It can be shown that the corresponding solutions of the MOP have uniform spread and also weak Pareto optimal points. It is notable that the NBI and proposed methods are independent of the relative scale of different objective functions. However, it is quite possible that obtaining a solution of the NBI method not be Pareto optimal (not even locally). Actually, this method aims at getting boundary points rather than Pareto optimal points that will lead to these points which may or may not be a Pareto optimal point. The effectiveness of this method is demonstrated with various test problems in convex and non-convex MOP cases. After that, a few test instances of the CEC 2009 (Zhang et al. 2008) using the proposed method are studied. Also, the relationship between the optimal solutions of the scalarized problem and the Pareto solutions of the multi-objective optimization problem is presented by several theorems.

On convex projective manifolds and cusps

This study of properly or strictly convex real projective manifolds introduces notions of parabolic, horosphere and cusp. Results include a Margulis lemma and in the strictly convex case a thick–thin decomposition. Finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds. This is proved using a characterization of ellipsoids in projective space.Except in dimension 3, there are only finitely many topological types of strictly convex manifolds with bounded volume. In dimension 4 and higher, the diameter of a closed strictly convex manifold is at most 9 times the diameter of the thick part. There is an algebraic characterization of strict convexity in terms of relative hyperbolicity.

Total diameter and area of closed submanifolds

The total diameter of a closed planar curve $$C\subset \mathbf {R}^2$$ is the integral of its antipodal chord lengths. We show that this quantity is bounded below by twice the area of $$C$$ . Furthermore, when $$C$$ is convex or centrally symmetric, the lower bound is twice as large. Both inequalities are sharp and the equality holds in the convex case only when $$C$$ is a circle. We also generalize these results to $$m$$ dimensional submanifolds of $$\mathbf {R}^n$$ , where the “area” will be defined in terms of the mod $$2$$ winding numbers of the submanifold about the $$n-m-1$$ dimensional affine subspaces of $$\mathbf {R}^n$$ .

Nonconvex cases for carpenter's rulers

We consider the carpenter's ruler folding problem in the plane, i.e., finding a minimum area shape with diameter 1 that accommodates foldings of any ruler whose longest link has length 1. An upper bound of π/3−3/4=0.614… and a lower bound of 10+25/8=0.475… are known for convex cases. We generalize the problem to simple nonconvex cases: in this setting we improve the upper bound to 0.583 and establish the first lower bound of 0.073. A variation is to consider rulers with at most k links. The current best convex upper bounds are (about) 0.486 for k=3,4 and π/6=0.523… for k=5,6. These bounds also apply to nonconvex cases. We derive a better nonconvex upper bound of 0.296 for k=3,4.

Accelerated gradient methods for nonconvex nonlinear and stochastic programming

In this paper, we generalize the well-known Nesterov's accelerated gradient (AG) method, originally designed for convex smooth optimization, to solve nonconvex and possibly stochastic optimization problems. We demonstrate that by properly specifying the stepsize policy, the AG method exhibits the best known rate of convergence for solving general nonconvex smooth optimization problems by using first-order information, similarly to the gradient descent method. We then consider an important class of composite optimization problems and show that the AG method can solve them uniformly, i.e., by using the same aggressive stepsize policy as in the convex case, even if the problem turns out to be nonconvex. We demonstrate that the AG method exhibits an optimal rate of convergence if the composite problem is convex, and improves the best known rate of convergence if the problem is nonconvex. Based on the AG method, we also present new nonconvex stochastic approximation methods and show that they can improve a few existing rates of convergence for nonconvex stochastic optimization. To the best of our knowledge, this is the first time that the convergence of the AG method has been established for solving nonconvex nonlinear programming in the literature.

Computing the Rate Distortion Region for the CEO Problem With Independent Sources

A method for numerically calculating the rate distortion region for the central estimation officer (CEO) problem when the sources are independent is derived by generalizing the Blahut-Arimoto algorithm. Unlike the traditional rate distortion function computation problem, the Lagrangian for the CEO rate distortion region can be nonconvex. When the Lagrangian is convex, the presented algorithm is converged from every initialization to the global optimum under some additional uniqueness conditions. When the Lagrangian is nonconvex, the convergent value obtained by the algorithm can be initialization dependent. To handle these nonconvex cases, an explicit nonrandom initialization that is in the region of attraction of the global optimum for low distortions is provided. Some example problems motivated by remote lossy function computation in sensor networks and wireless resource controllers highlight that both the convex and nonconvex cases occur in practice and the utility of the algorithm in computing their rate distortion regions.

Continuous dependence property of BSDE with constraints

In this paper, we study continuous properties of adapted solutions for backward stochastic differential equations with constraints (CBSDEs in short). Comparing with many existing literatures about this topic, our case is very general in the sense that constraints are formulated by general non-negative real functions. In general case, we proved a continuous property from below and a lower semi-continuous property of the minimal super-solution of CBSDE in its effective domain. Furthermore, in the special convex case, we obtained a continuous property with the help of convex analysis.

Existence results for Caputo type sequential fractional differential inclusions with nonlocal integral boundary conditions

In this paper, we study a new class of Caputo type sequential fractional differential inclusions with nonlocal Riemann–Liouville fractional integral boundary conditions. The existence of solutions for the given problem is established for the cases of convex and non-convex multivalued maps by using standard fixed point theorems. The obtained results are well illustrated with the aid of examples.

A Computationally Efficient FPTAS for Convex Stochastic Dynamic Programs

We propose a computationally efficient fully polynomial-time approximation scheme (FPTAS) to compute an approximation with arbitrary precision of the value function of convex stochastic dynamic programs, using the technique of $K$-approximation sets and functions introduced by Halman et al. [Math. Oper. Res., 34, (2009), pp. 674--685]. This paper deals with the convex case only, and it has the following contributions. First, we improve on the worst-case running time given by Halman et al. Second, we design and implement an FPTAS with excellent computational performance and show that it is faster than an exact algorithm even for small problem instances and small approximation factors, becoming orders of magnitude faster as the problem size increases. Third, we show that with careful algorithm design, the errors introduced by floating point computations can be bounded, so that we can provide a guarantee on the approximation factor over an exact infinite-precision solution. We provide an extensive computational evaluation based on randomly generated problem instances coming from applications in supply chain management and finance. The running time of the FPTAS is both theoretically and experimentally linear in the size of the uncertainty set.

A Nonsmooth Approach to Envelope Theorems

We develop a nonsmooth approach to envelope theorems applicable to a broad class of parameterized constrained nonlinear optimization problems that arise typically in economic applications with nonconvexities and/or nonsmooth objectives. Our methods emphasize the role of the Strict Mangasarian-Fromowitz Constraint Qualification (SMFCQ), and include envelope theorems for both the convex and nonconvex case, allow for noninterior solutions as well as equality and inequality constraints. We give new sufficient conditions for the value function to be directionally differentiable, as well as continuously differentiable. We apply our results to stochastic growth models with Markov shocks and constrained lattice programming problems.

大動脈炎に合併した円蓋部肥厚性硬膜炎の1例

大動脈炎に合併した円蓋部肥厚性硬膜炎の1例

A Solver for Nonconvex Bound-Constrained Quadratic Optimization

We present a new algorithm for nonconvex bound-constrained quadratic optimization. In the strictly convex case, our method is equivalent to the state-of-the-art algorithm by Dostál and Schöberl [Comput. Optim. Appl., 30 (2005), pp. 23--43]. Unlike their method, however, we establish a convergence theory for our algorithm that holds even when the problems are nonconvex. This is achieved by carefully addressing the challenges associated with directions of negative curvature, in particular, those that may naturally arise when applying the conjugate gradient algorithm to an indefinite system of equations. Our presentation and analysis deal explicitly with both lower and upper bounds on the optimization variables, whereas the work by Dostál and Schöberl considers only strictly convex problems with lower bounds. To handle this generality, we introduce the reduced chopped gradient that is analogous to the reduced free gradient previously used. The reduced chopped gradient leads to a new condition that is used to determine when optimization over a given subspace should be terminated. This condition, although not equivalent, is motivated by a similar condition used by Dostál and Schöberl. Numerical results illustrate the superior performance of our method over commonly used solvers that employ gradient projection steps and subspace acceleration.

Entropy generation based approach on natural convection in enclosures with concave/convex side walls

Computational study of natural convection within differentially heated enclosures with curved (concave/convex) side walls is carried out via entropy generation analysis. Numerical simulation has been carried out for various Prandtl numbers (Pr=0.015 and 1000) and Rayleigh numbers (103⩽Ra⩽105) with different wall curvatures. Results are presented in terms of isotherms (θ), streamlines (ψ), entropy generation due to heat transfer (Sθ) and fluid friction (Sψ). The effects of Rayleigh number on the total entropy generation (Stotal), average Bejan number (Beav) and global heat transfer rate (Nu‾r) are examined for all the cases. Maximum values of Sθ (Sθ,max) are found at the middle portion of the side walls for concave cases, whereas, Sθ,max is observed near the top right and bottom left corner of the cavity for convex cases. On the other hand, Sψ,max is seen near the solid walls of the cavity for all concave and convex cases. At all Ra and low Pr, largest heat transfer rate and lesser entropy generation is found for case 3 (highly concave case). Overall, for convex case, case 1 or case 2 (lesser convex cases) are efficient for all Ra and Pr. On the other hand, case 3 of concave case (highly concave) offers larger heat transfer rate and lesser entropy generation compared to less concave and all convex cases at low Ra and all Pr. At high Ra and low Pr, case 3 (concave) may be the optimal case whereas, at high Ra and high Pr, case 1 (less concave) may be recommended based on higher heat transfer rate. A comparative study of the concave and convex cases also revealed that the concave cases with high concavity (case 3) may be chosen as the energy efficient case at high Ra and high Pr.

Worst case complexity of direct search under convexity

In this paper we prove that the broad class of direct-search methods of directional type, based on imposing sufficient decrease to accept new iterates, exhibits the same worst case complexity bound and global rate of the gradient method for the unconstrained minimization of a convex and smooth function. More precisely, it will be shown that the number of iterations needed to reduce the norm of the gradient of the objective function below a certain threshold is at most proportional to the inverse of the threshold. It will be also shown that the absolute error in the function values decay at a sublinear rate proportional to the inverse of the iteration counter. In addition, we prove that the sequence of absolute errors of function values and iterates converges r-linearly in the strongly convex case.