Nonconvex Relaxation Research Articles

Restricted $p$-Isometry Properties of Nonconvex Matrix Recovery

Recently, a nonconvex relaxation of low-rank matrix recovery (LMR), called the Schatten- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> quasi-norm minimization (0 <; <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> <; 1), was introduced instead of the previous nuclear norm minimization in order to approximate the problem of LMR closer. In this paper, we introduce a notion of the restricted <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -isometry constants (0 <; <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> ≤ 1) and derive a <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -RIP condition for exact reconstruction of LMR via Schatten- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> quasi-norm minimization. In particular, we determine how many random, Gaussian measurements are needed for the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -RIP condition to hold with high probability, which gives a theoretical result that it needs fewer measurements with small <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> for exact recovery via Schatten- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> quasi-norm minimization than when <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> =1.

Restricted Normal Cones and Sparsity Optimization with Affine Constraints

The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely behaved nonconvex relaxations. In this paper we consider the elementary method of alternating projections (MAP) for solving the sparsity optimization problem without employing convex heuristics. In a parallel paper we recently introduced the restricted normal cone which generalizes the classical Mordukhovich normal cone and reconciles some fundamental gaps in the theory of sufficient conditions for local linear convergence of the MAP algorithm. We use the restricted normal cone together with the notion of superregularity, which is inherently satisfied for the affine sparse optimization problem, to obtain local linear convergence results with estimates for the radius of convergence of the MAP algorithm applied to sparsity optimization with an affine constraint.

A decomposition approach for a new test-scenario in complex problem solving

Over the last years, psychological research has increasingly used computer-supported tests, especially in the analysis of complex human decision making and problem solving. The approach is to use computer-based test scenarios and to evaluate the performance of participants and correlate it to certain attributes, such as the participant's capacity to regulate emotions. However, two important questions can only be answered with the help of modern optimization methodology. The first one considers an analysis of the exact situations and decisions that led to a bad or good overall performance of test persons. The second important question concerns performance, as the choices made by humans can only be compared to one another, but not to the optimal solution, as it is unknown in general.Additionally, these test-scenarios have usually been defined on a trial-and-error basis, until certain characteristics became apparent. The more complex models become, the more likely it is that unforeseen and unwanted characteristics emerge in studies. To overcome this important problem, we propose to use mathematical optimization methodology not only as an analysis and training tool, but also in the design stage of the complex problem scenario.We present a novel test scenario, the IWR Tailorshop, with functional relations and model parameters that have been formulated based on optimization results. We also present a tailored decomposition approach to solve the resulting mixed-integer nonlinear programs with nonconvex relaxations and show some promising results of this approach.

A Feasible Nonconvex Relaxation Approach to Feature Selection

Variable selection problems are typically addressed under apenalized optimization framework. Nonconvex penalties such as the minimax concave plus (MCP) and smoothly clipped absolute deviation(SCAD), have been demonstrated to have the properties of sparsity practically and theoretically. In this paper we propose a new nonconvex penalty that we call exponential-type penalty. The exponential-type penalty is characterized by a positive parameter,which establishes a connection with the ell0 and ell1 penalties.We apply this new penalty to sparse supervised learning problems. To solve to resulting optimization problem, we resort to a reweighted ell1 minimization method. Moreover, we devise an efficient method for the adaptive update of the tuning parameter. Our experimental results are encouraging. They show that the exponential-type penalty is competitive with MCP and SCAD.

Optimization strategies for discrete multi-material stiffness optimization

Design of composite laminated lay-ups are formulated as discrete multi-material selection problems. The design problem can be modeled as a non-convex mixed-integer optimization problem. Such problems are in general only solvable to global optimality for small to moderate sized problems. To attack larger problem instances we formulate convex and non-convex continuous relaxations which can be solved using gradient based optimization algorithms. The convex relaxation yields a lower bound on the attainable performance. The optimal solution to the convex relaxation is used as a starting guess in a continuation approach where the convex relaxation is changed to a non-convex relaxation by introduction of a quadratic penalty constraint whereby intermediate-valued designs are prevented. The minimum compliance, mass constrained multiple load case problem is formulated and solved for a number of examples which numerically confirm the sought properties of the new scheme in terms of convergence to a discrete solution.

Global optimization of truss topology with discrete bar areas—Part II: Implementation and numerical results

A classical problem within the field of structural optimization is to find the stiffest truss design subject to a given external static load and a bound on the total volume. The design variables describe the cross sectional areas of the bars. This class of problems is well-studied for continuous bar areas. We consider here the difficult situation that the truss must be built from pre-produced bars with given areas. This paper together with Part I proposes an algorithmic framework for the calculation of a global optimizer of the underlying non-convex mixed integer design problem. In this paper we use the theory developed in Part I to design a convergent nonlinear branch-and-bound method tailored to solve large-scale instances of the original discrete problem. The problem formulation and the needed theoretical results from Part I are repeated such that this paper is self-contained. We focus on the implementation details but also establish finite convergence of the branch-and-bound method. The algorithm is based on solving a sequence of continuous non-convex relaxations which can be formulated as quadratic programs according to the theory in Part I. The quadratic programs to be treated within the branch-and-bound search all have the same feasible set and differ from each other only in the objective function. This is one reason for making the resulting branch-and-bound method very efficient. The paper closes with several large-scale numerical examples. These examples are, to the knowledge of the authors, by far the largest discrete topology design problems solved by means of global optimization.

Truss topology optimization with discrete design variables—Guaranteed global optimality and benchmark examples

This paper considers the problem of optimal truss topology design subject to multiple loading conditions. We minimize a weighted average of the compliances subject to a volume constraint. Based on the ground structure approach, the cross-sectional areas are chosen as the design variables. While this problem is well-studied for continuous bar areas, we consider in this study the case of discrete areas. This problem is of major practical relevance if the truss must be built from pre-produced bars with given areas. As a special case, we consider the design problem for a single available bar area, i.e., a 0/1 problem. In contrast to the heuristic methods considered in many other approaches, our goal is to compute guaranteed globally optimal structures. This is done by a branch-and-bound method for which convergence can be proven. In this branch-and-bound framework, lower bounds of the optimal objective function values are calculated by treating a sequence of continuous but non-convex relaxations of the original mixed-integer problem. The main effect of using this approach lies in the fact that these relaxed problems can be equivalently reformulated as convex problems and, thus, can be solved to global optimality. In addition, these convex problems can be further relaxed to quadratic programs for which very efficient numerical solution procedures exist. By exploiting this special problem structure, much larger problem instances can be solved to global optimality compared to similar mixed-integer problems. The main intention of this paper is to provide optimal solutions for single and multiple load benchmark examples, which can be used for testing and validating other methods or heuristics for the treatment of this discrete topology design problem.

Non-Convex Relaxation to the Space of Hypercomplex Numbers for Indefinite Quadratic Programming-Properties and Optimization Algorithms

This paper addresses a sort of non-convex relaxation problem for the minimization problem of a quadratic function. We define relaxation problems by generalizing the feasible region of the original problem to the space consisting of hypercomplex numbers. Computational experiments for 0-1 quadratic minimization problems reveal the effectiveness of two proposed algorithms based on the derived properties. Fundamental properties of the relaxation problem for a more general class of quadratic minimization problems are also discussed.

0-1 Optimization of Quadratic Form by Expanding the Parameter Space

This paper is concerned with the boolean quadratic optimization problem. We formulate and analyze a class of non-convex relaxation problems which includes the relaxation problem with complex variables and the SDP relaxation problem as special cases. The effects of expanding the parameter space of decision variables to a space of hypercomplex number are investigated. It is shown that for any instance of problem data the relaxation problem in complex variable is the strongest non-convex relaxation among the relaxations (in our formulation) under the condition of having “monotonically decreasing path” which connects any two feasible solutions of the original problem.

Convergence rates to travelling waves for a nonconvex relaxation model

In this paper we study the asymptotic behaviour of the solution for a nonconvex relaxation model. The time decay rates in both the exponential and algebraic forms of the travelling wave solutions are shown by the weighted energy method. Our results develop and improve the stability theory in [8,9].

Stability of a Traffic Flow Model with Nonconvex Relaxation

This paper is concerned with the nonlinear stability of traveling wave solutions for a quasi-linear relaxation model with a nonconvex equilibrium flux. The study is motivated by and the results are applied to the well-known dynamic continuum traffic flow model, the Payne and Whitham (PW) model with a nonconcave fundamental diagram. The PW model is the first of its kind and it has been widely adopted by traffic engineers in the study of stability and instability phenomena of traffic flow. The traveling wave solutions are shown to be asymptotically stable under small disturbances and under the sub-characteristic condition using a weighted energy method. The analysis applies to both non-degenerate case and the degenerate case where the traveling wave has exponential decay rates at infinity and has an algebraic decay rate at infinity, respectively.