Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSOS-convex Semi-algebraic Programs and its Applications to Robust Optimization: A Tractable Class of Nonsmooth Convex Optimization
184
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we introduce a new class of nonsmooth convex functions called SOS-convex semialgebraic functions extending the recently proposed notion of SOS-convex polynomials. This class of nonsmooth convex functions covers many common nonsmooth functions arising in the applications such as the Euclidean norm, the maximum eigenvalue function and the least squares functions with $ell_1$-regularization or elastic net regularization used in statistics and compressed sensing. We show that, under commonly used strict feasibility conditions, the optimal value and an optimal solution of SOS-convex semi-algebraic programs can be found by solving a single semi-definite programming problem (SDP). We achieve the results by using tools from semi-algebraic geometry, convex-concave minimax theorem and a recently established Jensen inequality type result for SOS-convex polynomials. As an application, we outline how the derived results can be applied to show that robust SOS-convex optimization problems under restricted spectrahedron data uncertainty enjoy exact SDP relaxations. This extends the existing exact SDP relaxation result for restricted ellipsoidal data uncertainty and answers the open questions left in [Optimization Letters 9, 1-18(2015)] on how to recover a robust solution from the semi-definite programming relaxation in this broader setting.
rate research
Read More
This paper presents a convex sufficient condition for solving a system of nonlinear equations under parametric changes and proposes a sequential convex optimization method for solving robust optimization problems with nonlinear equality constraints. By bounding the nonlinearity with concave envelopes and using Brouwers fixed point theorem, the sufficient condition is expressed in terms of closed-form convex inequality constraints. We extend the result to provide a convex sufficient condition for feasibility under bounded uncertainty. Using these conditions, a non-convex optimization problem can be solved as a sequence of convex optimization problems, with feasibility and robustness guarantees. We present a detailed analysis of the performance and complexity of the proposed condition. The examples in polynomial optimization and nonlinear network are provided to illustrate the proposed method.
Minimax optimization problems are an important class of optimization problems arising from modern machine learning and traditional research areas. While there have been many numerical algorithms for solving smooth convex-concave minimax problems, numerical algorithms for nonsmooth convex-concave minimax problems are very rare. This paper aims to develop an efficient numerical algorithm for a structured nonsmooth convex-concave minimax problem. A majorized semi-proximal alternating coordinate method (mspACM) is proposed, in which a majorized quadratic convex-concave function is adopted for approximating the smooth part of the objective function and semi-proximal terms are added in each subproblem. This construction enables the subproblems at each iteration are solvable and even easily solved when the semiproximal terms are cleverly chosen. We prove the global convergence of the algorithm mspACM under mild assumptions, without requiring strong convexity-concavity condition. Under the locally metrical subregularity of the solution mapping, we prove that the algorithm mspACM has the linear rate of convergence. Preliminary numerical results are reported to verify the efficiency of the algorithm mspACM.
The paper proposes and justifies a new algorithm of the proximal Newton type to solve a broad class of nonsmooth composite convex optimization problems without strong convexity assumptions. Based on advanced notions and techniques of variational analysis, we establish implementable results on the global convergence of the proposed algorithm as well as its local convergence with superlinear and quadratic rates. For certain structural problems, the obtained local convergence conditions do not require the local Lipschitz continuity of the corresponding Hessian mappings that is a crucial assumption used in the literature to ensure a superlinear convergence of other algorithms of the proximal Newton type. The conducted numerical experiments of solving the $l_1$ regularized logistic regression model illustrate the possibility of applying the proposed algorithm to deal with practically important problems.
Hidden convex optimization is such a class of nonconvex optimization problems that can be globally solved in polynomial time via equivalent convex programming reformulations. In this paper, we focus on checking local optimality in hidden convex optimization. We first introduce a class of hidden convex optimization problems by jointing the classical nonconvex trust-region subproblem (TRS) with convex optimization (CO), and then present a comprehensive study on local optimality conditions. In order to guarantee the existence of a necessary and sufficient condition for local optimality, we need more restrictive assumptions. To our surprise, while (TRS) has at most one local non-global minimizer and (CO) has no local non-global minimizer, their joint problem could have more than one local non-global minimizer.
We optimize a general model of bioprocesses, which is nonconvex due to the microbial growth in the biochemical reactors. We formulate a convex relaxation and give conditions guaranteeing its exactness in both the transient and steady state cases. When the growth kinetics are modeled by the Monod function under constant biomass or the Contois function, the relaxation is a second-order cone program, which can be solved efficiently at large scales. We implement the model on a numerical example based on a wastewater treatment system.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
