Concave Objective Research Articles

Convergence Analysis of Quantized Primal-Dual Algorithms in Network Utility Maximization Problems

This paper investigates the asymptotic and nonasymptotic behavior of the quantized primal-dual (PD) algorithm in network utility maximization (NUM) problems, in which a group of agents maximizes the sum of their individual concave objective functions under linear constraints. In the asymptotic scenario, we use the information-theoretic notion of differential entropy power to establish universal bounds on the maximum exponential convergence rates of joint PD, primal and dual variables under optimum-achieving quantization schemes. These results provide tradeoffs between the speed of exponential convergence, the agents' objective functions, the communication bit rates, and the number of agents and constraints. In the nonasymptotic scenario, we obtain lower bounds on the mean square distance of joint PD, primal and dual variables from the optimal solution at any time instant. These bounds hold regardless of the quantization scheme used.

Light Interreflections and Shadowing Effects in a Lambertian V-Cavity under Diffuse Illumination

The different areas of a concave object illuminate each other by a multiple light reflection process, called interreflections, depending on the geometries of the object and the lighting. For an accurate prediction of the radiance perceived from each point of the object by an observer or a camera, an interreflection model is necessary, taking into account the optical properties and the shape of the object, the orientation(s) of the incident light which can produce shadows, and the infinite number of light bounces between the different points of the object. The present paper focusses on the irradiance of two adjacent planar panels (V-cavity) illuminated by collimated light from any direction of the hemisphere, or by diffuse light. According to the reflectance of the material and the angle of the cavity, the loss of irradiance near the fold due to the shadowing effect is partly compensated by the gain in radiance due to the interreflections.

A new exact algorithm for concave knapsack problems with integer variables

ABSTRACTConcave knapsack problems with integer variables have many applications in real life, and they are NP-hard. In this paper, an exact and efficient algorithm is presented for concave knapsack problems. The algorithm combines the contour cut with a special cut to improve the lower bound and reduce the duality gap gradually in the iterative process. The lower bound of the problem is obtained by solving a linear underestimation problem. A special cut is performed by exploiting the structures of the objective function and the feasible region of the primal problem. The optimal solution can be found in a finite number of iterations, and numerical experiments are also reported for two different types of concave objective functions. The computational results show the algorithm is efficient.

Interreflections in Computer Vision: A Survey and an Introduction to Spectral Infinite-Bounce Model

Interreflections are observed on concave objects or when multiple objects are located closely. In a vision system, interreflections can largely affect color values captured by the camera. Due to this fact, modeling interreflections is important for many vision applications. In this paper, we consider the problem of treating and modeling interreflections in the domain of computer vision. First, a survey of existing approaches in the state of the art is given. These approaches are detailed, discussed and compared. Most of the state of the art models take into consideration only two bounces of light between surface elements. We, afterward, introduce a new interreflection model based on radiometric definitions. This model is the first one that takes into consideration an infinite number of light bounces between surface elements while providing image RGB values as a result. The accuracy of our model is studied by comparing it to real camera outputs. Thanks to our new model, the importance of using infinite bounces of light while studying interreflection, instead of only two bounces, is demonstrated.

On the Construction of Optimal Payoffs

The quantile formulation for optimal portfolio selection problems under increasing law-invariant objectives allows to reduce any such problem to an optimization problem on real functions under monotonicity conditions. We solve two basic types of these optimization problems, which makes it possible to solve in a unified way several portfolio selection problems of interest. In particular, we completely solve the optimal portfolio selection problem for an investor with preferences as in Yaari's dual theory of choice. Extending previous work we also derive a reduction result in general form when the payoff is required to have a fixed copula with some benchmark (state-dependent constraint). Specifically, we show that if one can determine the optimal payoff under a concave law-invariant objective, then one can also determine the optimal payoff when adding the state-dependent constraint. Finally, we identify market conditions which ensure attainability of the optimal payoff by means of a static portfolio of puts and calls.

Refining Transient Electromagnetic Scattering Analysis: A new approach based on the magnetic field integral equation

The magnetic field integral equation (MFIE) in a time-domain form is used to describe transient electromagnetic (EM) scattering by conductors. The MFIE is a second kind of integral equation that can help improve the conditioning of the matrix equation but is less widely addressed in the time domain. The time-domain MFIE (TDMFIE) is generally solved by the spatial-domain method of moments (MoM) at each time step, after being discretized in the time domain with a march-on-in-time (MOT) scheme. The MoM is inconvenient in implementation due to its special basis and testing functions, and, in addition, the MOT will produce the well-known instability problem. This article proposes a novel solving approach by incorporating a spatial-domain Nystr?m scheme with a robust singularity treatment that has a temporal-domain Galerkin method (TDGM) with Laguerre basis and testing functions. The Nystr?m scheme has several advantages, as demonstrated in the frequency domain, but it has not been used for the TDMFIE. The TDGM can fully overcome the drawback of the MOT and simplify the implementation by deriving closed-form formulations. Numerical examples for transient EM scattering by conductors including a concave object are presented here to illustrate the approach and its robust results.

Contact Force Decomposition Using Contact Pressure Distribution

Many robotics applications involving contact with objects require the decomposition of an external force measured by a force sensor into its normal and shear components. In this paper, we present a general and effective contact force decomposition method based on the contact pressure distribution. The contact pressure distribution can be measured easily using recently developed flexible pressure sensor arrays. Our method requires no preprocessing and provides accurate force decomposition, even for concave external objects. These advantages are demonstrated by simulations and experiments.

Calibration of Distributionally Robust Empirical Optimization Models

We study the out-of-sample properties of robust empirical optimization problems with smooth φ-divergence penalties and smooth concave objective functions, and develop a theory for data-driven calibration of the non-negative “robustness parameter” δ that controls the size of the deviations from the nominal model. Building on the intuition that robust optimization reduces the sensitivity of the expected reward to errors in the model by controlling the spread of the reward distribution, we show that the first-order benefit of “little bit of robustness” (i.e., δ small, positive) is a significant reduction in the variance of the out-of-sample reward while the corresponding impact on the mean is almost an order of magnitude smaller. One implication is that substantial variance (sensitivity) reduction is possible at little cost if the robustness parameter is properly calibrated. To this end, we introduce the notion of a robust mean-variance frontier to select the robustness parameter and show that it can be approximated using resampling methods like the bootstrap. Our examples show that robust solutions resulting from “open loop” calibration methods (e.g., selecting a 90% confidence level regardless of the data and objective function) can be very conservative out- of-sample, while those corresponding to the robustness parameter that optimizes an estimate of the out-of-sample expected reward (e.g., via the bootstrap) with no regard for the variance are often insufficiently robust.

A fast continuous method for the extreme eigenvalue problem

Matrix eigenvalue problems play a significant role in many areas of computational science and engineering. In this paper, we propose a fast continuous method for the extreme eigenvalue problem. We first convert such a nonconvex optimization problem into a minimization problem with concave objective function and convex constraints based on the continuous methods developed by Golub and Liao. Then, we propose a continuous method for solving such a minimization problem. To accelerate the convergence of this method, a self-adaptive step length strategy is adopted. Under mild conditions, we prove the global convergence of this method. Some preliminary numerical results are presented to verify the effectiveness of the proposed method eventually.

Optimal Mechanisms for Combinatorial Auctions and Combinatorial Public Projects via Convex Rounding

We design the first truthful-in-expectation, constant-factor approximation mechanisms for NP -hard cases of the welfare maximization problem in combinatorial auctions with nonidentical items and in combinatorial public projects. Our results apply to bidders with valuations that are nonnegative linear combinations of gross-substitute valuations, a class that encompasses many of the most well-studied subclasses of submodular functions, including coverage functions and weighted matroid rank functions. Our mechanisms have an expected polynomial runtime and achieve an approximation factor of 1 − 1/ e . This approximation factor is the best possible for both problems, even for known and explicitly given coverage valuations, assuming P ≠ NP . Recent impossibility results suggest that our results cannot be extended to a significantly larger valuation class. Both of our mechanisms are instantiations of a new framework for designing approximation mechanisms based on randomized rounding algorithms. The high-level idea of this framework is to optimize directly over the (random) output of the rounding algorithm , rather than the usual (and rarely truthful) approach of optimizing over the input to the rounding algorithm. This framework yields truthful-in-expectation mechanisms, which can be implemented efficiently when the corresponding objective function is concave. For bidders with valuations in the cone generated by gross-substitute valuations, we give novel randomized rounding algorithms that lead to both a concave objective function and a (1 − 1/ e )-approximation of the optimal welfare.

Seeing more clearly through psychosis: Depth inversion illusions are normal in bipolar disorder but reduced in schizophrenia

Schizophrenia patients with more positive symptoms are less susceptible to depth inversion illusions (DIIs) in which concave objects appear as convex. It remains unclear, however, the extent to which this perceptual advantage uniquely characterizes the schizophrenia phenotype. To address the foregoing, we compared 30 bipolar disorder patients to a previously published sample of healthy controls (N=25) and schizophrenia patients (N=30). The task in all cases was to judge the apparent convexity of physically concave faces and scenes. Half of the concave objects were painted with realistic texture to enhance the convexity illusion and the remaining objects were untextured to reduce the illusion. Subjects viewed objects stereoscopically or via monocular motion parallax depth cues. For each group, DIIs were stronger with texture than without, and weaker with stereoscopic information than without, indicating a uniformly normal response to stimulus alterations across groups. Bipolar patients experienced DIIs more frequently than schizophrenia patients but as commonly as controls, irrespective of the face/scene category, texture, or viewing condition (motion/stereo). More severe positive and disorganized symptoms predicted reduced DIIs for schizophrenia patients and across all patients. These results suggest that people with schizophrenia, but not bipolar disorder, more accurately perceive object depth structure. Psychotic symptoms—or their accompanying neural dysfunction—may primarily drive the effect presumably through eroding the visual system's generalized tendency to construe unusual or ambiguous surfaces as convex. Because such symptoms are by definition more common in schizophrenia, DIIs are at once state-sensitive and diagnostically specific, offering a potential biomarker for the presence of acute psychosis.

Full-matrix capture with phased shift migration for flaw detection in layered objects with complex geometry

This paper introduces a method for an ultrasonic imaging with a phased array based on a wave migration algorithm. The method allows for imaging layered objects with lateral velocity variations such as objects with a complex geometry or layers that are not perpendicular to the array’s axis. The full-matrix capture ensures that there is enough information to reconstruct an image even when the wave indication angle is large. The method is implemented in a omega-k domain.The proposed algorithm is first tested in a single simulation of a concave object with side drilled holes under the concave surface. For evaluating the algorithm’s performance three experiments are presented: one with a tilted object (surface not perpendicular with respect to the array axis) with side drilled holes and two experiments of an object with concave surface and two artificial defects under it. The results presented in the paper verify that the proposed method reconstructs images from the data gathered with the phased array.

Function approximation for building the algorithm for the rough terrain description

The paper studies one version of the piecewise polynomial approximation using the "possible directions" method and G. Zoutendijk's method to solve the problems of describing complex functions. In particular, the problem with one quadratic constraint was presented and the methods of quadratic programming with a prior statement of the dual problems were used to solve it. To solve this problem we use an approach based on the duality theory applying a direct algorithm of the simplex method. The algorithm is presented with the goal of the further software implementation. The conclusions are made about the practical value of this research, in particular, about the possibility of expanding the tools for decision-makers for describing the affected areas of rough terrain by man-made accidents and justification of a new approach for constructing three-dimensional models of convex and concave objects.

Analyzing process flexibility: A distribution-free approach with partial expectations

We develop a distribution-free model to evaluate the performance of process flexibility structures when only the mean and partial expectation of the demand are known. We characterize the worst-case demand distribution under general concave objective functions, and apply it to derive tight lower bounds for the performance of chaining structures under the balanced systems (systems with the same number of plants and products). We also derive a simple lower bound for chaining-like structures under unbalanced systems with different plant capacities.

Analyzing Process Flexibility: A Distribution-Free Approach with Partial Expectations

We develop a distribution-free model to evaluate the performance of process flexibility structures when only the mean and partial expectation of the demand are known. We characterize the worst-case demand distribution under general concave objective functions, and applying it to derive tight lower bounds for the performance of chaining structures under the balanced systems (systems with the same number of plants and products). We also derive a simple lower bound for chaining-like structures under unbalanced systems with different plant capacities.

Assessment of trypophobia and an analysis of its visual precipitation.

We developed and validated a symptom scale that can be used to identify "trypophobia", in which individuals experience aversion induced by images of clusters of circular objects. The trypophobia questionnaire (TQ) was based on reports of various symptom types, but it nevertheless demonstrated a single construct, with high internal consistency and test-retest reliability. The TQ scores predicted discomfort from trypophobic images, but not neutral or unpleasant images, and did not correlate with anxiety. Using image filtering, we also reduced the excess energy at midrange spatial frequencies associated with both trypophobic and uncomfortable images. Relative to unfiltered trypophobic images, the discomfort from filtered images experienced by observers with high TQ scores was less than that experienced with control images and by observers with low TQ scores. Furthermore, we found that clusters of concave objects (holes) did not induce significantly more discomfort than clusters of convex objects (bumps), suggesting that trypophobia involves images with particular spectral profile rather than clusters of holes per se.

Entrepreneurial Decisions on Effort and Project with a Nonconcave Objective Function

We propose and solve a general entrepreneurial/managerial decision-making problem. Instead of employing concave objective functions, we use a broad class of nonconcave objective functions. We approach the problem by a martingale method. We show that the optimization problem with a nonconcave objective function has the same solution as the optimization problem when the objective function is replaced by its concave hull, and thus the problems are equivalent to each other. The value function is shown to be strictly concave and to satisfy the Hamilton-Jacobi-Bellman equation of dynamic programming. We also show that the final wealth cannot take values in the region where the objective function is not concave: the entrepreneur would like to avoid her or his wealth ending up in the nonconcave region. Because of this, the entrepreneur’s risk taking explodes as time nears maturity if her his wealth is equal to the right end point of the nonconcave region.

A SmartPen for 3D interaction and sketch-based surface modeling

This paper presents an innovative, portable, and low cost system for fast and interactive design of sketched models using 3D curve networks. The system is based on a stereo camera and a wireless pen-shaped device, called the SmartPen, which can be extended or bent to facilitate concave objects’ acquisitions. Its tip position is tracked in 3D and used to trace the style lines of the object to be modeled in the virtual space. Starting from the 3D irregular curve network, progressively drawn by the user, a new Sketch-based Modeling Framework automatically creates the inferred 3D virtual model. The interactive system integrates perceptual and conceptual modeling, allowing both for the reconstruction of existing objects via user-drawn strokes on the physical object and for free-hand strokes which together contribute to realize the desired 3D curve network. Real-time reconstruction of the sketched surface interpolating the 3D curve network is obtained through an interactive curvature-based optimization approach which determines the smooth subdivision surface that best fits the traced 3D curve network.

A Neurodynamic Approach for Real-Time Scheduling via Maximizing Piecewise Linear Utility.

In this paper, we study a set of real-time scheduling problems whose objectives can be expressed as piecewise linear utility functions. This model has very wide applications in scheduling-related problems, such as mixed criticality, response time minimization, and tardiness analysis. Approximation schemes and matrix vectorization techniques are applied to transform scheduling problems into linear constraint optimization with a piecewise linear and concave objective; thus, a neural network-based optimization method can be adopted to solve such scheduling problems efficiently. This neural network model has a parallel structure, and can also be implemented on circuits, on which the converging time can be significantly limited to meet real-time requirements. Examples are provided to illustrate how to solve the optimization problem and to form a schedule. An approximation ratio bound of 0.5 is further provided. Experimental studies on a large number of randomly generated sets suggest that our algorithm is optimal when the set is nonoverloaded, and outperforms existing typical scheduling strategies when there is overload. Moreover, the number of steps for finding an approximate solution remains at the same level when the size of the problem (number of jobs within a set) increases.

Envelope theorems in Banach lattices and asset pricing

We develop envelope theorems for optimization problems in which the value function takes values in a general Banach lattice. We consider both the special case of a convex choice set and a concave objective function and the more general case case of an arbitrary choice set and a general objective function. We apply our results to discuss the existence of a well-defined notion of marginal utility of wealth in optimal discrete-time, finite-horizon consumption-portfolio problems with an unrestricted information structure and preferences allowed to display habit formation and state dependency.