In this paper, the strictly convex quadratic program (QP) arising in model predictive control for constrained linear systems is reformulated as a system of piecewise affine equations. A regularized piecewise smooth Newton method with exact line-search on a convex, differentiable, piecewise-quadratic merit function is proposed for the solution of the reformulated problem. The algorithm has consi...

We test the efficiency of two pairwise comparison methods. The methods studied are Salminen’s piecewise linear prospect theory (PLP) method and the convex cone method by Korhonen, Wallenius, and Zionts (KWZ). The PLP method is based on a piecewise linear difference function. The KWZ method assumes a quasi-concave utility or value function. These methods are tested using randomly generated data ...

Semi-supervised Support vector machine has become an increasingly popular tool for machine learning due to its wide applicability. Unlike SVM, their formulation leads to a non-smooth non-convex optimization problem. In 2005, Chapelle and Zien used a Gaussian approximation as a smooth function and presented ∇TSVM. In this paper, we propose a smooth piecewise function and research smooth piecewis...

The α-reformulation (αR) technique can be used to transform any nonconvex twice-differentiable mixedinteger nonlinear programming problem to a convex relaxed form. By adding a quadratic function to the nonconvex function it is possible to convexify it, and by subtracting a piecewise linearization of the added function a convex underestimator will be obtained. This reformulation technique is imp...

We consider the problem of fitting a continuous piecewise linear function to a finite set of data points, modeled as a mathematical program with convex objective. We review some fitting problems that can be modeled as convex programs, and then introduce mixed-binary generalizations that allow variability in the regions defining the best-fit function’s domain. We also study the additional constr...

We present a novel layerwise optimization algorithm for the learning objective of Piecewise-Linear Convolutional Neural Networks (PL-CNNs), a large class of convolutional neural networks. Specifically, PL-CNNs employ piecewise linear non-linearities such as the commonly used ReLU and max-pool, and an SVM classifier as the final layer. The key observation of our approach is that the problem corr...

We utilize support functions to transform the problem of constructing the convex hull of a finite set of curved objects into the problem of computing the upper envelope of piecewise linear functions. This approach is particularly suited if the objects are (possibly intersecting) circular arcs in the plane. Mathematics Subject Classification (2000). Primary 68U05; Secondary 65D18.

We study mixed-integer nonlinear optimization (MINLO) formulations of the disjunction $$x\in \{0\}\cup [\ell ,u]$$ , where z is a binary indicator for ( $$0 \le \ell <u$$ ), and y “captures” f(x), which assumed to be convex positive on its domain $$[\ell but otherwise $$y=0$$ when $$x=0$$ . This model very useful in combinatorial optimization, there fixed cost c operating an activity at level x...

Despite strong connections through shared application areas, research efforts on power market optimization (e.g., unit commitment) and network optimal flow) remain largely independent. A notable illustration of this is the treatment generation cost functions, where nonlinear has used polynomial representations adopted piecewise linear encodings. This work combines state-of-the-art results from ...