can smt/z3 be used for optimazation - z3

Question

Can SMT solver efficiently find a solution (or an assignment) for the pseudo-Boolean problem as described as follows:
\sum {i..m} f_i x1 x2.. xn *w_i
where f_i x1 x2 .. xn is a Boolean function, and w_i is a weight of Int type.
For your convenience, I highlight the contents in page 1 and 3, which is enough for specifying
the pseudo-Boolean problem.

Answer 1

SMT solvers typically address the question: given a logical formula, optionally using functions and predicates from underlying theories (such as the theory of arithmetic, the theory of bit-vectors, arrays), is the formula satisfiable or not.
They typically don't expose a way for you specify objective functions
and typically don't have built-in optimization procedures.
Some special cases are formulas that only use Booleans or a combination of Booleans and either bit-vectors or integers. Pseudo Boolean constraints can be formulated with either integers or encoded (with some care taking overflow semantics into account) using bit-vectors, or they can be encoded directly into SAT. For some formulas using bounded integers that fall in the class of psuedo-boolean problems, Z3 will try automatic reductions into bit-vectors. This applies only to benchmkars in the SMT-LIB2 format tagged as QF_LIA or applies if you explicitly invoke a tactic that performs this reduction (the "qflia" tactic should apply).
While Z3 does not directly expose objective functions, the question of augmenting
SMT solvers with objective functions is actively pursued in the research community.
One approach suggested by Nieuwenhuis and Oliveras in SAT 2006 was to build in
solving for the "weighted max SMT" problem as a custom theory. Yices comes with built-in
features for weighted max SMT, Z3 does not, but it is possible to write a custom
theory that performs the backtracking search of a weighted max SMT solver, but nothing
out of the box.
Sometimes people try to specify objective functions using quantified formulas.
In theory one could hope that quantifier elimination procedures then can solve
for the objective.
This is generally pretty bad when it comes to performance. Quantifier elimination
is an overfit and the routines (that we have) will not be efficient.

Answer 2

For your problem, if you want to find an optimized (maximum or minimum) result from the sum, yes Z3 has this ability. You can use the Optimize class of Z3 library instead of Solver class. The class provides two methods for 'maximization' and 'minimization' respectively. You can pass the SMT variable that is needed to be optimized and Optimization class model will give the solution for you. It actually worked with C# API using Microsoft.Z3 library. For your inconvenience, I am attaching a snippet:
Optimize opt; // initializing object
opt.MkMaximize(*your variable*);
opt.MkMinimize(*your variable*);
opt.Assert(*anything you need to do*);