Using the dual cone of sums of nonnegative circuits (SONC), we provide a relaxation of the global optimization problem to minimize an exponential sum and, as a special case, a multivariate real polynomial. Our approach builds on two key observations. First, that the dual SONC cone is contained in the primal one. Hence, containment in this cone is a certificate of nonnegativity. Second, we show that membership in the dual cone can be verified by a linear program. We implement the algorithm and present initial experimental results comparing our method to existing approaches.
The cone of sums of nonnegative circuits (SONCs) is a subset of the cone of nonnegative polynomials / exponential sums, which has been studied extensively in recent years. In this article, we construct a subset of the SONC cone which we call the DSONC cone. The DSONC cone can be seen as an extension of the dual SONC cone; membership can be tested via linear programming. We show that the DSONC cone is a proper, full-dimensional cone, we provide a description of its extreme rays, and collect several properties that parallel those of the SONC cone. Moreover, we show that functions in the DSONC co…
We introduce and study a cone which consists of a class of generalized polynomial functions and which provides a common framework for recent non-negativity certificates of polynomials in sparse settings. Specifically, this $\mathcal{S}$-cone generalizes and unifies sums of arithmetic-geometric mean exponentials (SAGE) and sums of non-negative circuit polynomials (SONC). We provide a comprehensive characterization of the dual cone of the $\mathcal{S}$-cone, which even for its specializations provides novel and projection-free descriptions. As applications of this result, we give an exact charac…
We provide a complete and explicit characterization of the real zeros of sums of nonnegative circuit (SONC) polynomials, a recent certificate for nonnegative polynomials independent of sums of squares. As a consequence, we derive an exact determination of the number $B''_{n+1,2d}$ for all $n$ and $d$. $B''_{n+1,2d}$ is defined to be the supremum of the number of zeros of all homogeneous $n+1$-variate polynomials of degree $2d$ in the SONC cone. The analogously defined numbers $B_{n+1,2d}$ and $B'_{n+1,2d}$ for the nonnegativity cone and the cone of sums of squares were first introduced and stu…
Finding the minimum of a multivariate real polynomial is a well-known hard problem with various applications. We present a polynomial time algorithm to approximate such lower bounds via sums of nonnegative circuit polynomials (SONC). As a main result, we carry out the first large-scale comparison of SONC, using this algorithm and different geometric programming (GP) solvers, with the classical sums of squares (SOS) approach, using several of the most common semidefinite programming (SDP) solvers. SONC yields bounds competitive to SOS in several cases, but using significantly less time and memo…
Recently, the second and the third author developed sums of nonnegative circuit polynomials (SONC) as a new certificate of nonnegativity for real polynomials, which is independent of sums of squares. In this article we show that the SONC cone is full-dimensional in the cone of nonnegative polynomials. We establish a Positivstellensatz which guarantees that every polynomial which is positive on a given compact, semi-algebraic set can be represented by the constraints of the set and SONC polynomials. Based on this Positivstellensatz we provide a hierarchy of lower bounds converging against the m…
For a non-empty, finite subset $\mathcal{A} \subseteq \mathbb{N}_0^n$, denote by $C_{\text{sonc}}(\mathcal{A}) \in \mathbb{R}[x_1, \ldots, x_n]$ the cone of sums of non-negative circuit polynomials with support $\mathcal{A}$. We derive a representation of the dual cone $(C_{\text{sonc}}(\mathcal{A}))^*$ and deduce a resulting optimality criterion for the use of sums of non-negative circuit polynomials in polynomial optimization.
Circuit polynomials are polynomials satisfying a number of conditions that make it easy to compute sharp and certifiable global lower bounds for them. Consequently, one may use them to find certifiable lower bounds for any polynomial by writing it as a sum of circuit polynomials with known lower bounds (if possible), in a fashion similar to the better-known sum-of-squares polynomials. Seidler and de Wolff recently showed that sums of nonnegative circuit (SONC) polynomials can be used to compute global lower bounds (called SONC bounds) for polynomials in this manner in polynomial time, as long …
The $\mathcal{S}$-cone provides a common framework for cones of polynomials or exponential sums which establish non-negativity upon the arithmetic-geometric inequality, in particular for sums of non-negative circuit polynomials (SONC) or sums of arithmetic-geometric exponentials (SAGE). In this paper, we study the $\mathcal{S}$-cone and its dual from the viewpoint of second-order representability. Extending results of Averkov and of Wang and Magron on the primal SONC cone, we provide explicit generalized second-order descriptions for rational $\mathcal{S}$-cones and theirs duals.
The second-order cone is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit a representation using second-order cones, given that they have a strong expressive ability. In this paper, we prove constructively that the cone of sums of nonnegative circuits (SONC) admits a second-order cone representation. Based on this, we give a new algorithm to compute SONC decompositions for certain classes of nonnegative polynomials via second-order con…
We provide two hybrid numeric-symbolic optimization algorithms, computing exact sums of nonnegative circuits (SONC) and sums of arithmetic-geometric-exponentials (SAGE) decompositions. Moreover, we provide a hybrid numeric-symbolic decision algorithm for polynomials lying in the interior of the SAGE cone. Each framework, inspired by previous contributions of Parrilo and Peyrl, is a rounding-projection procedure. For a polynomial lying in the interior of the SAGE cone, we prove that the decision algorithm terminates within a number of arithmetic operations, which is polynomial in the number of …
Recently, Chubanov proposed an interesting new polynomial-time algorithm for linear program. In this paper, we extend his algorithm to second-order cone programming.
Kuhn-Tucker conditions for mathematical programming problems in Banach spaces partially ordered by cone with empty interior are obtained under strong simultaneity condition. If partial ordered cone has interior point, it is proved that Slater and strong simultaneity conditions are equivalent.
Motivated by the expressive power of completely positive programming to encode hard optimization problems, many approximation schemes for the completely positive cone have been proposed and successfully used. Most schemes are based on outer approximations, with the only inner approximations available being linear programming based methods proposed by Bundfuss and Dür and also Yıldırım, and a semidefinite programming based method proposed by Lasserre. In this paper, we propose the use of the cone of nonnegative scaled diagonally dominant matrices as a natural inner approximation to the complete…
We introduce a conic embedding condition that gives a hierarchy of cones and cone programs. This condition is satisfied by a large number of convex cones including the cone of copositive matrices, the cone of completely positive matrices, and all symmetric cones. We discuss properties of the intermediate cones and conic programs in the hierarchy. In particular, we demonstrate how this embedding condition gives rise to a family of cone programs that interpolates between LP, SOCP, and SDP. This family of $k$th order cones may be realized either as cones of $n$-by-$n$ symmetric matrices or as con…
This paper addresses the compliance minimization of a truss, where the number of available nodes is limited. It is shown that this optimization problem can be recast as a second-order cone programming with a cardinality constraint. We propose a simple heuristic based on the alternative direction method of multipliers. The efficiency of the proposed method is compared with a global optimization approach based on mixed-integer second-order cone programming. Numerical experiments demonstrate that the proposed method often finds a solution having a good objective value with small computational cos…
Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, %% on these polynomials, we prove that the optimality conditions always hold on optimizers, and the coordinates of optimizers are algebraic functions of the coefficients of the input polynomials. We also give a general formula for the algebraic degree of the optimal coordinates. The derivation of the algebraic degree is equivalent to counting the number of all complex critical points. As special cases, we obtain the algebraic degrees of …
This paper is devoted to the study of tilt stability of local minimizers, which plays an important role in both theoretical and numerical aspects of optimization. This notion has been comprehensively investigated in the unconstrained framework as well as for problems of nonlinear programming with $C^2$-smooth data. Available results for nonpolyhedral conic programs were obtained only under strong constraint nondegeneracy assumptions. Here we develop an approach of second-order variational analysis, which allows us to establish complete neighborhood and pointbased characterizations of tilt stab…
The constant rank constraint qualification, introduced by Janin in 1984 for nonlinear programming, has been extensively used for sensitivity analysis, global convergence of first- and second-order algorithms, and for computing the derivative of the value function. In this paper we discuss naive extensions of constant rank-type constraint qualifications to second-order cone programming and semidefinite programming, which are based on the Approximate-Karush-Kuhn-Tucker necessary optimality condition and on the application of the reduction approach. Our definitions are strictly weaker than Robins…
Second-order necessary optimality conditions for nonlinear conic programming problems that depend on a single Lagrange multiplier are usually built under nondegeneracy and strict complementarity. In this paper we establish a condition of such type for two classes of nonlinear conic problems, namely semidefinite and second-order cone programming, assuming Robinson's constraint qualification and a weak constant rank-type property which are, together, strictly weaker than nondegeneracy. Our approach is done via a penalty-based strategy, which is aimed at providing strong global convergence result…
In this paper the simplicial cone constrained convex quadratic programming problem is studied. The optimality conditions of this problem consist in a linear complementarity problem. This fact, under a suitable condition, leads to an equivalence between the simplicial cone constrained convex quadratic programming problem and the one of finding the unique solution of a nonsmooth system of equations. It is shown that a semi-smooth Newton method applied to this nonsmooth system of equations is always well defined and under a mild assumption on the simplicial cone the method generates a sequence th…
We propose an interior point method (IPM) for solving semidefinite programming problems (SDPs). The standard interior point algorithms used to solve SDPs work in the space of positive semidefinite matrices. Contrary to that the proposed algorithm works in the cone of matrices of constant \emph{factor width}. This adaptation makes the proposed method more suitable for parallelization than the standard IPM. We prove global convergence and provide a complexity analysis. Our work is inspired by a series of papers by Ahmadi, Dash, Majumdar and Hall, and builds upon a recent preprint by Roig-Solvas …
Among many approaches to increase the computational efficiency of semidefinite programming (SDP) relaxation for quadratic constrained quadratic programming problems (QCQPs), exploiting the aggregate sparsity of the data matrices in the SDP by Fukuda et al. (2001) and second-order cone programming (SOCP) relaxation have been popular. In this paper, we exploit the aggregate sparsity of SOCP relaxation of QCQPs. Specifically, we prove that exploiting the aggregate sparsity reduces the number of second-order cones in the SOCP relaxation, and that we can simplify the matrix completion procedure by …
The paper conducts a second-order variational analysis for an important class of nonpolyhedral conic programs generated by the so-called second-order/Lorentz/ice-cream cone $Q$. From one hand, we prove that the indicator function of $Q$ is always twice epi-differentiable and apply this result to characterizing the uniqueness of Lagrange multipliers at stationary points together with an error bound estimate in the general second-order cone setting involving ${\cal C}^2$-smooth data. On the other hand, we precisely calculate the graphical derivative of the normal cone mapping to $Q$ under the we…
The objective of this work is to study weak infeasibility in second order cone programming. For this purpose, we consider a relaxation sequence of feasibility problems that mostly preserve the feasibility status of the original problem. This is used to show that for a given weakly infeasible problem at most $m$ directions are needed to approach the cone, where $m$ is the number of Lorentz cones. We also tackle a closely related question and show that given a bounded optimization problem satisfying Slater's condition, we may transform it into another problem that has the same optimal value but …
In [R. Andreani, G. Haeser, L. M. Mito, H. Ramírez C., Weak notions of nondegeneracy in nonlinear semidefinite programming, arXiv:2012.14810, 2020] the classical notion of nondegeneracy (or transversality) and Robinson's constraint qualification have been revisited in the context of nonlinear semidefinite programming exploiting the structure of the problem, namely, its eigendecomposition. This allows formulating the conditions equivalently in terms of (positive) linear independence of significantly smaller sets of vectors. In this paper we extend these ideas to the context of nonlinear second-…
This paper studies siting and sizing of plug-in electric vehicle (PEV) fast-charging stations on coupled transportation and power networks. We develop a closed-form service rate model of highway PEV charging stations' service abilities, which considers heterogeneous PEV driving ranges and charging demands.We utilize a modified capacitated flow refueling location model (CFRLM) to explicitly capture time-varying PEV charging demands on the transportation network under driving range constraints. We explore extra constraints of the CFRLM to enhance model accuracy and computational efficiency.We th…
The optimal control problem for the kinematic bicycle model is considered where the trajectories are required to satisfy the safety constraints in the continuous-time sense. Based on the differential flatness property of the model, necessary and sufficient conditions in the flat space are provided to guarantee safety in the state space. The optimal control problem is relaxed to the problem of solving three second-order cone programs (SOCPs) sequentially, which find the safe path, the trajectory duration, and the speed profile, respectively. Solutions of the three SOCPs together provide a sub-o…
Low levels of system inertia in power grids with significant penetration of non-synchronous Renewable Energy Sources (RES) have increased the risk of frequency instability. The provision of a certain type of ancillary services such as inertia and Frequency Response (FR) is needed at all times, to maintain system frequency within secure limits if the loss of a large power infeed were to occur. In this paper we propose a frequency-secured optimisation framework for the procurement of inertia and FR with diverse dynamics, which enables to apply a marginal-pricing scheme for these services. This p…
This paper approximates simulation models by B-splines with a penalty on high-order finite differences of the coefficients of adjacent B-splines. The penalty prevents overfitting. The simulation output is assumed to be nonnegative. The nonnegative spline simulation metamodel is casted as a second-order cone programming model, which can be solved efficiently by modern optimization techniques. The method is implemented in MATLAB/GNU Octave.
This paper studies the worst case iteration complexity of an infeasible interior point method (IPM) for seconder order cone programming (SOCP), which is more convenient for warmstarting compared with feasible IPMs. The method studied bases on the homogeneous and self-dual model and the Monteiro-Zhang family of searching directions. Its worst case iteration complexity is $O\left(k^{1/2}\log\left(ε^{-1}\right)\right)$, to reduce the primal residual, dual residual, and complementarity gap by a factor of $ε$, where $k$ is the number of cone constraints. The result is the same as the best known res…
We develop a natural variant of Dikin's affine-scaling method, first for semidefinite programming and then for hyperbolic programming in general. We match the best complexity bounds known for interior-point methods. All previous polynomial-time affine-scaling algorithms have been for conic optimization problems in which the underlying cone is symmetric. Hyperbolicity cones, however, need not be symmetric. Our algorithm is the first polynomial-time affine-scaling method not relying on symmetry.
An important problem in tree breeding is optimal selection from candidate pedigree members to produce the highest performance in seed orchards, while conserving essential genetic diversity. The most beneficial members should contribute as much as possible, but such selection of orchard parents would reduce performance of the orchard progeny due to serious inbreeding. To avoid inbreeding, we should include a constraint on the numerator relationship matrix to keep a group coancestry under an appropriate threshold. Though an SDP (semidefinite programming) approach proposed by Pong-Wong and Woolli…
The well known constant rank constraint qualification [Math. Program. Study 21:110--126, 1984] introduced by Janin for nonlinear programming has been recently extended to a conic context by exploiting the eigenvector structure of the problem. In this paper we propose a more general and geometric approach for defining a new extension of this condition to the conic context. The main advantage of our approach is that we are able to recast the strong second-order properties of the constant rank condition in a conic context. In particular, we obtain a second-order necessary optimality condition tha…
In this paper, we present new convex relaxations for nonconvex quadratically constrained quadratic programming (QCQP) problems. While recent research has focused on strengthening convex relaxations using reformulation-linearization technique (RLT), the state-of-the-art methods lose their effectiveness when dealing with (multiple) nonconvex quadratic constraints in QCQP. In this research, we decompose and relax each nonconvex constraint to two second order cone (SOC) constraints and then linearize the products of the SOC constraints and linear constraints to construct some effective new valid c…
In order to express a polyhedron as the (Minkowski) sum of a polytope and a polyhedral cone, Motzkin (1936) made a transition from the polyhedron to a polyhedral cone. Based on his excellent idea, we represent a set by a characteristic cone. By using this representation, we then reach four main results: (i) expressing a closed convex set containing no line as the direct sum of the convex hull of its extreme points and conical hull of its extreme directions, (ii) establishing a convex programming (CP) based framework for determining a maximal element-an element with the maximum number of positi…
We study the problem of designing attacks to safety-critical systems in which the adversary seeks to maximize the overall system cost within a model predictive control framework. Although in general this problem is NP-hard, we characterize a family of problems that can be solved in polynomial time via a second-order cone programming relaxation. In particular, we show that positive systems fall under this family. We provide examples demonstrating the design of optimal attacks on an autonomous vehicle and a microgrid.
Hyperbolic programming is the problem of computing the infimum of a linear function when restricted to the hyperbolicity cone of a hyperbolic polynomial, a generalization of semidefinite programming. We propose an approach based on symbolic computation, relying on the multiplicity structure of the algebraic boundary of the cone, without the assumption of determinantal representability. This allows us to design exact algorithms able to certify the multiplicity of the solution and the optimal value of the linear function.
We develop a decomposition algorithm for distributionally-robust two-stage stochastic mixed-integer convex cone programs, and its important special case of distributionally-robust two-stage stochastic mixed-integer second order cone programs. This generalizes the algorithm proposed by Sen and Sherali~[Mathematical Programming 106(2): 203-223, 2006]. We show that the proposed algorithm is finitely convergent if the second-stage problems are solved to optimality at incumbent first stage solutions, and solution to an optimization problem to identify worst-case probability distribution is availabl…
Nonlinear Convex Cone Programming (NCCP) problems are important and have many practical applications. In this paper, we introduces a flexible first-order primal-dual algorithm called the Variant Auxiliary Problem Principle (VAPP) for solving NCCP problems when the objective function and constraints are smooth and may be nonsmooth. Each iteration of VAPP generates a nonlinear approximation to the primal problem of an augmented Lagrangian method. The approximation incorporates both linearization and a variable distance-like function, and then the iterations of VAPP provide one decomposition prop…
The goal of this paper is to investigate new and simple convergence analysis of dynamic programming for linear quadratic regulator problem of discrete-time linear time-invariant systems. In particular, bounds on errors are given in terms of both matrix inequalities and matrix norm. Under a mild assumption on the initial parameter, we prove that the Q-value iteration exponentially converges to the optimal solution. Moreover, a global asymptotic convergence is also presented. These results are then extended to the policy iteration. We prove that in contrast to the Q-value iteration, the policy i…
