Dose Escalation Study of MLN0128 in Combination With Paclitaxel, With/Without Trastuzumab, in Subjects With Advanced Solid Malignancies

Inclusion Criteria:

* Voluntary written consent
* Locally advanced or metastatic solid tumors with the exception of primary brain tumor, and have failed or are not eligible for standard of care therapy
* Eastern Cooperative Oncology Group (ECOG) performance status 0-1
* Ability to swallow oral medications
* For women of child-bearing potential, negative serum or urine pregnancy test within 14 days prior to the first study drug administration and use of physician-approved method of birth control from 30 days prior to 30 days following the last study drug administration
* Male participants must be surgically sterile or must agree to use physician-approved contraception during the study and for 30 days following the last study drug administration
* Clinical laboratory values as specified in the protocol
* For expansion phase (Arm A) - HER2-/unknown participants will be enrolled
* For expansion phase (Arm B) - HER2+ cancer participants will be enrolled

Exclusion Criteria:

* Diagnosis of primary brain tumor
* Have received prior cancer or other investigational therapy within 2 weeks prior to the first administration of study drug
* Known impaired cardiac function or clinically significant cardiac disease
* Known treatment with systemic corticosteroid within one week prior to the first administration of study drug
* Diabetes mellitus
* Human immunodeficiency virus (HIV) infection
* Known active cardiovascular disease condition as specified in protocol
* Pregnancy (positive serum or urine pregnancy test) or breast feeding
* Malabsorption due to prior gastrointestinal (GI) surgery, GI disease
* Other clinically significant co-morbidities

Please note that there are additional inclusion and exclusion criteria. The study center will determine if you meet all of the criteria.

Site personnel will explain the trial in detail and answer any question you may have if you do qualify for the study. You can then decide whether or not you wish to participate. If you do not qualify for the trial, site personnel will explain the reasons